HANDOUT 6 Flashcards
Why is omitting a relevant variable an issue?
BIASED coefficients
E(b1) when we omit a relevant variable
E(b1) = B1 + B2 [COV(X, Z)/Var(X)] ≠ B1
2 things the bias is determined by
- Sign of the coefficient on the omitted variable
2. The covariance between the included and omitted variables
Can we test for an omitted relevant variable? Why/why not?
NO - Zi is unknown.
if we knew the omitted relevant variable, we’d have included it.
Taylor series expansion formula
f(x) = f(x0) + f’(x0)(x - x0)/1! + f’‘(x0)(x - x0)^2/2!
+ … + f^n(x0)(x - x0)^n / n!
What does a taylor series expansion allow us to do?
We can always approximate a non-linear function as a high order polynomial
How does a taylor series expansion change the error term in the true model?
Ei –> Vi + remainder
Since the taylor expansion is only an approximation.
For a reset test, since Ui (false error term) is unobserved, what do we use?
Ui hat = residuals
Reset Test form & explain
Ui hat = d1 yi^2 hat + d2 yi^3 hat + … + [r0 + r1X1i + … + rk Xki]
- We use the fitted valyes yi^2 hat as a proxy for x1^2,…,xn^2, x1x1,…,xn-1 xn
- Include intercept & original set of X variables to get correct DOF for unrestricted
H0 and H1 for reset test
H0: d1 = d2 = 0 - functional form OK
H1: dj≠0 - functional form WRONG
What test statistic do we use for reset test?
F = [(RSS R - RSS U)/d] / [RSS U / dof]
What is our restricted model?
The ‘false’ model that we thought the functional form may be wrong
Problem with reset test and why
LOW POWERED TEST
- As yi^ hats only a proxy
- Means we often do not reject H0 even tho we should = often find that the functional form is ok when it isn’t.
How does inclusion of irrelevant variables affect unbiasedness?
It doesn’t - estimates still unbiased.
E(b1) = B1
Why is inclusion of irrelevant variables an issue?
In V(b1) the denominator = RSS when in the true (bivariate) model it's TSS. RSS < TSS. So including the irrelevant variable means we get too big variances = big SE = small t-ratios = often do NOT reject H0 & often find insignificant coefficients when they're not.
