Why can autocorrelation occur?

-Inertia

-Specification, excluding a variable

-Lag, regressor dependent on previous period

-Manipulation / smoothing of data

-Data transformation

Show teh Autoregressive 1 ( AR(1) ) model?

x

Show the Moving Average ( MA(1) ) model

x

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

-Durbin-Watson test

-Breusch-Godfrey test

How is the durbin watson test calculated?

x

What does the Durbin-Watson test assume?

-Regression has intercept

-Nonstochastic Xi's

-AR(1)

-Normal errors

-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))

-1

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 d_{L }& d_{U} from tables

if d L or 4-d_{L} < d -> then autocorrelation

if d_{U }< d < 4-D_{U} then no autocorrelation

Otherwise undecisive

D_{U} & D_{L} are dependent on n & k-1

How is the Breusch-Godfrey test run?

1. Run the normal regression, may include lagged regressants

2. Regress u_{t} on Xi, ρ_{1}u_{t-1} ρ_{2}u_{t-p }other Xi's

3. For large samples, nR^{2} ~ 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 = ρu_{t-1 }+ ϵ

We can tranform our model by -ρY_{t-1}

So Y_{t} - ρY_{t-1 }= B_{1}(1-ρ)+B_{2}(X_{t}-ρX_{t-1}) + ϵ_{t}

or Y_{i}*=B_{1}*+B_{2}*xt*+ ϵ_{t}

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 u_{t}

2. Then run the model u_{t} = ρ_{1}u_{t-1} + v_{t}

3. use ρ_{1 }to use the tranformed model, get new residuals

4. Use the new residuals to resestimate ρ_{1}

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 = B_{1} + B_{2}X_{2t}+u_{t}

u_{t} _{~ }N(0, α_{0} + α_{1} u^{2}_{t-1})

run u^{2}_{t ~ }α_{0}+ α_{2}u^{2}_{t-1} ... α_{k}u^{2}_{k-x}

Use nR^{2 }_{~} Chi(k)