Topic 11: Autocorrelation

Question 1

Why can autocorrelation occur?

Answer 1

• Inertia
• Specification, excluding a variable
• Lag, regressor dependent on previous period
• Manipulation / smoothing of data
• Data transformation

Question 2

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

Answer 2

x

Question 3

Show the Moving Average ( MA(1) ) model

Answer 3

x

Question 4

How does uncorrected Autocorrelation affect our results?

Answer 4

• Estimators still linear & unbiased, not BLUE
• No minimum variance
• Variance ill estimated, likely under
• Tests not valid

Question 5

What can be done to test for autocorrelation?

Answer 5

• Graph of u^i v time
• Durbin-Watson test
• Breusch-Godfrey test

Question 6

How is the durbin watson test calculated?

Answer 6

x

Question 7

What does the Durbin-Watson test assume?

Answer 7

• Regression has intercept
• Nonstochastic Xi’s
• AR(1)
• Normal errors
• no lagged regressant in the model
• No missing observations

Question 8

How is rho related to the d stat in the DW test?

Answer 8

d ~= (2(1-ρ))

-1 < ρ < 1 so 0 <= d <= 4

When d = 0, p = 1.

When d = 2, ρ = 0

When d = 4, ρ = -1

Question 9

How does one check a durbin watson d stat?

Answer 9

Look up d_L & d_U from tables

if d < 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

Question 10

How is the Breusch-Godfrey test run?

Answer 10

1. Run the normal regression, may include lagged regressants
2. Regress u_t on Xi, ρ₁u_t-1 ρ₂u_t-pother 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

Question 11

How might we correct AR(1) for known ρ?

Answer 11

Because ut = ρu_t-1 + ϵ

We can tranform our model by -ρY_t-1

So Y_t - ρY_t-1 = B₁(1-ρ)+B₂(X_t-ρX_t-1) + ϵ_t

or Y_i*=B₁*+B₂*xt*+ ϵ_t

Coefficients are now blue

We must remember to adjust coefficients for interpretation

Question 12

How can we use the durbin watson test to estimate ρ?

Answer 12

ρ = 1 - d/2

Question 13

What is the Cochrane-Orcut procedure?

Answer 13

1. Run the normal model, get u_t
2. Then run the model u_t = ρ₁u_t-1 + v_t
3. use ρ₁to 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

Question 14

What are the Newey-West errors?

Answer 14

Like whites errors, but for autocorrelation. Not BLUE but valid tests

Question 15

What is ARCH?

Answer 15

Autoregressive Conditional Heteroscedasticity Model

Yt = B₁ + B₂X_2t+u_t

u_{t ~} N(0, α₀ + α₁ u²_t-1)

run u²_{t ~}α₀+ α₂u²_t-1 … α_ku²_k-x

Use nR²_~ Chi(k)