Grading scale: 80+ A, 70-79 B, -69 C. Special guest readings will be extra credit.
Serial Correlation:
-properties of OLS with serially correlated errors
-OLS standard errors and tests invalid
-OlS will not be efficient
-OLS still be unbiased and consistant
Example: \(y_t=a+bx_{1t}+cx_{2t}+\epsilon\)
This is the restricted model above
\(y_t=a_1+b_1x_{1t}+c_1x_{2t}+\epsilon\)
\(y_t=a_2+b_2x_{1t}+c_2x_{2t}+\epsilon\)
This is the unrestricted model above
The basic test used is the chow test: \(\frac{(SSR_R -(S_1+S_2)/k}{(S_1+S_2)/N_1+N_2-2k)}\)
Serial Correlation and the pretense of lagged variables
-including too few lags will cause an omitted variable problem and serial correlation
-why? you want too many generally because ommitting a variable can throw off estimates themselves.
Testing for serial correlation
- \(y_t=B_0 +B_1 x_{t1}+...+B_k x_{tk}+u_t\)
- \(u_t=e u_{t-1}+e_t\)
- get the residuals: \(\hat{u_t}\)
-What are we doing? Run the OLS regression, use the x's and y's to attain the ut hats. Then run the second equation to get rho hat, then run OLS on them. To check serial correlation by using t-test with rho=0 as the null.
Durbin-Watson Test AR(1)
\(DW=\frac{\Sigma^n _{t=2} (\hat{u_t}-\hat{u_{t-1}})^2}{\Sigma^n _{t=2} u^2_t} \approx 2(1-\hat{\rho})\)
\(H_0: \rho=0\)
Reject if: \(DW<d_L\)
fail to reject if: \(DW>d_u\)
\(DW=.8<d_L=1.32\) in book example.
What if regressors are not fixed or exogenous?
Testing for AR(1) with serial correlation and general regressors
\(\hat{u_t}=\alpha_0+\alpha_1 x_{1t} +...+\alpha x{tk}+\rho \hat{u_{t-1}}+error\)
Again, testing \(H_0: \rho=0\)
Want to know if variables are endogenous, run F test.
Breush-Godfry test for AR(a)
\(\hat{u_t}=\alpha_0+\alpha_1 x_{1t} +...+\alpha_k x{tk}+\rho \hat{u_{t-1}}+...+e_q \hat{u_{t-q}}\)
Correcting for serial correlation with exogenous regressors
\(y_t=B_0+B_1 x_t+u\) transformed into
\(\rho y_t=\rho B_0+\rho B_1 x_t+u\)
\(y_t-\rho y_{t-1}=B_0(1-\rho)+B_1 (x_t-\rho x_{t-1}) +(ut-\rho u_{t-1})\)
Newey-West
-coreelation factors
-uses correlation factors for correction
Heteroskedasticity in time series regressions
-heteroskedasticity and autocorrelation constitent models
-heteroskedasticity receives less attention than serial correlation in time series
-fixing heteroskedasticity but not serial correlation doesn't help you much
Autoregressive Conditional Heteroskedasticity (ARCH)
- \(VAR (u_t |x,u_{t-1},....u_{t-k})=\alpha_0 +\alpha_1 u^2_{t-1}\)
If there are no lagged dependent variables among regressors OLS remains BLUE