Week 4 - Classical Assumption Violations Flashcards
Classical assumptions are unrealistic
What do we remove from CLRA (2)
IN TIME SERIES: We now say errors are autocorellated (Against CLRA4)
IN CROSS SECTION: We now find errors are heteroscedastic (σ² varies) (Against CLRA5)
As well as autocorrelated errors, we should consider autocorrelated variables
What does it mean if a variable is autocorrelated
(in time series models)
It is correlated with itself at different points in time
Autocorrelated variables expression:
Xt= pXt-1 + εt
Variable X at time t is a function of itself in the last period.
P is a parameter called the autocorrelation coefficient.
Example of a regression model of current variables affecting future dependent variable
Yt+1 = β₀+β₁Xt+₁ +β₂Zt +εt+₁
Here we can see Zt (current Z) influences Yt+1 (future Y)
What happens when we include lags of variables in our models (2)
We lose observations so sample size N shrinks. (Smaller sample=less accurate)
More lags = more parameters (β) to estimate. so k increases too which lowers degrees of freedom n-(k+1), making less precise.
So both lower accuracy!
Covariance formula for:
Xt and its first lag Xt-1 expression
(Pg3 blue highlighter)
B) What happens to correlation as j gets bigger?
pσ² of x= pσ² of ε/ (1-p²)
Same as variance of current Xt but add p to σ².
For looking at cov (Xt,Xt-j), would be p to the j (the amount of lags) in the formula (above it would be p¹ since only looking at Xt and first lag Xt-1!)
B)
As j gets larger, further in past, and so correlation gets smaller, which makes sense!
(as results are likely to correlate with its near past rather than ages ago)
What do plots look like for different values of p
(p= autocorrelation coefficient) (pg4)
p=0 plot is random
p=0.9 strong pos, so smoother runs of positive and negative values
p=-0.9 strong neg auto, spikier plot
This was autocorrelation of variables, which does not violate the classical assumptions. What does violate the classical assumption
Autocorrelated ERRORS
So errors are autocorrelated (they are correlated with itself in different time periods)
Expression
εt = pεt-₁ + ut
p is autocorrelation coefficient
ut is error term assumed to satisfy the classical assumptions (important)
(Remember autocorrelated variable expression is
Xt = pXt-₁ + εt)
So using this expression, when are classical assumptions violated?
Classical assumption is violated unless p=0
(Since if p=0 it only leaves Ut which is an error term that does satisfy the terms)
Sources of error autocorrelation (2)
Omission of explanatory variables - if omitted variable is autocorrelated, error term is autocorrelated. (Known as false autocorrel since not due to the error itself!)
Dynamic structure
So omitting autocorrelated variables can mean errors are autocorrelated.
Why may they be in the first place omitted (2)
Do not realise significance
Not measureable. E.g ability So therefore the error term reflects it instead
If ommitted, it is contained in the ε. What does this look like as an example.
True model
𝑌t=𝛽₀ +𝛽₁𝑋₁ + β₂𝑋₂ + 𝜔t where 𝜔 is the disturbance.
If say 𝑋₁ is unobservable then the
model we estimate is
𝑌 =𝛽₀+𝛽₂𝑋₂ +𝜀t
And error is soaked up the X₁ meansthat 𝜀 =𝛽₁𝑋 ₁+𝜔t
2nd source of autocorrelated errors:
Dynamic structure
Not possible to model every factor influencing the dependent variable
