Why can autocorrelation occur?
-Specification, excluding a variable
-Lag, regressor dependent on previous period
-Manipulation / smoothing of data
Show teh Autoregressive 1 ( AR(1) ) model?
Show the Moving Average ( MA(1) ) model
How does uncorrected Autocorrelation affect our results?
-Estimators still linear & unbiased, not BLUE
-No minimum variance
-Variance ill estimated, likely under
-Tests not valid
What can be done to test for autocorrelation?
-Graph of u^i v time
How is the durbin watson test calculated?
What does the Durbin-Watson test assume?
-Regression has intercept
-no lagged regressant in the model
-No missing observations
How is rho related to the d stat in the DW test?
d ~= (2(1-ρ))
When d = 0, p = 1.
When d = 2, ρ = 0
When d = 4, ρ = -1
How does one check a durbin watson d stat?
Look up dL & dU from tables
if d L or 4-dL < d -> then autocorrelation
if dU < d < 4-DU then no autocorrelation
DU & DL are dependent on n & k-1
How is the Breusch-Godfrey test run?
1. Run the normal regression, may include lagged regressants
2. Regress ut on Xi, ρ1ut-1 ρ2ut-p other Xi's
3. For large samples, nR2 ~ Chi(p), or F(k,n-k-p-1)
- Works for MA,
- p must be assumed / guessed
- might want to choose a yearly p, so montly data would have p=12
How might we correct AR(1) for known ρ?
Because ut = ρut-1 + ϵ
We can tranform our model by -ρYt-1
So Yt - ρYt-1 = B1(1-ρ)+B2(Xt-ρXt-1) + ϵt
or Yi*=B1*+B2*xt*+ ϵt
Coefficients are now blue
We must remember to adjust coefficients for interpretation
How can we use the durbin watson test to estimate ρ?
ρ = 1 - d/2
What is the Cochrane-Orcut procedure?
1. Run the normal model, get ut
2. Then run the model ut = ρ1ut-1 + vt
3. use ρ1 to use the tranformed model, get new residuals
4. Use the new residuals to resestimate ρ1
5. Continue until ρ does not change much with each iteration
What are the Newey-West errors?
Like whites errors, but for autocorrelation. Not BLUE but valid tests
What is ARCH?
Autoregressive Conditional Heteroscedasticity Model
Yt = B1 + B2X2t+ut
ut ~ N(0, α0 + α1 u2t-1)
run u2t ~ α0+ α2u2t-1 ... αku2k-x
Use nR2 ~ Chi(k)