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March 31, 2005
Heteroskedasticity
Plotting the data or, especially, the residuals vs. one of the regressors is an effective test for heteroskedasticity. The Goldfeld-Quandt test formalizes the concept by looking at the residual variance for the largest third and the smallest third of the values for x.
A simple model of the process generating the heteroskedasticity leads to a simple transformation to homoskedasticity.
Another pattern (not heteroskasticity) might also be apparent from a plot of the residuals. Here we see a classic case of the wrong functional form.
Heteroskeasticity is really important in financial market research when it takes the form of Autoregressive Conditional Heteroskedasticity (ARCH). The prices of assets vary with their risk, which can be viewed as their conditional variance.
Posted by bparke at 08:58 PM | Comments (0)
March 29, 2005
HW
Due Tuesday 4/5: Chapter 12, #1-7.
Due Thursday 3/31: the following exercise that can be based largely on the do file mc2-rho.do
.
Consider 3 models:
1) y = a + b*x + e
2) y = a + b*x + u
3) y = a + b*x + c*y[-1] + e,
where u = rho*u[-1] + e and e is an i.i.d. normally distributed error term. You could estimate four models:
A) y = a + b*x + e (ignore possible serial correlation)
B) y = a + b*x + u (Cochrane-Orcutt, estimate rho)
C) y = a + b*x + c*y[-1] + e
D) y = a + c*y[-1] + e.
For all combinations of data generating processes (1, 2, 3) and estimated models (A, B,C, D), run simulations sufficient to understand the properties of the estimated models given each DGP. Make a table of some sort to illustrate your results (including dwstat).
For serious students: How do your results depend on the values of rho and a, b, and c?
Posted by bparke at 10:53 AM | Comments (0)
March 24, 2005
Serially Correlated Errors
Posted by bparke at 08:43 PM | Comments (0)
Slope and Intercept Dummy Variables
Slope and intercept dummies together effectively allow for two separate regression lines to explain two subsamples.
Leaving out the constant term constrains the regression line to run through the origin. This is seldom interesting.
Posted by bparke at 03:19 PM | Comments (0)
March 22, 2005
Serially Correlated Error Terms
We worked our way through some algebra that does not appear in the textbook. The first step was showing how the coefficient estimate is a function of the random error terms.
If the errors and regressors are not independent, you get bias in the estimate of beta.
The standard formula for the variance of a regression coefficient assumes that the errors are independent. If they are not, this formula leaves out some nonzero terms.
On the left, diagnosing serially correlated errors. In the middle, what happens if you have serially correlated errors. On the right, time series economic data is all serially correlated so it is easy to see why the errors from time series regressions might be as well.
The distribution of the Durbin-Watson statistic under the null of no serial correlation:
Side trip: One important application of the covariance is in determining the risk of a portfolio containing two assets.
Posted by bparke at 08:50 PM | Comments (0)
March 21, 2005
Dummy Variables
Exact collinearity: some linear combinations of the regressors (possibly including 1 for the intercept) is zero.
The problem with exact collinearity is that no amount of data can yield estimates of the parameters.
If there are three states, we could construct three state dummies. Adding all three to an equation with a constant term produces exact collinearity. The solution is either to add two of the three dummies or to suppress the constant term.
The Chow test is for the alternative hypothesis that a data sample could be broken up into m subsamples where the regression parameters differ across subsamples.
A closely related idea is to put in slope dummies and do an F test. This imposes the restriction that the error variance is the same across subsamples.
The choice among these two approaches sometimes comes down to computational convenience.
Posted by bparke at 10:37 PM | Comments (0)
March 08, 2005
Midterm 1 Answers
The pictures are most useful if you have a copy of the answers (printed handout). Being in class would also probably have been helpful.
Posted by bparke at 10:29 PM | Comments (0)
March 01, 2005
Structural Breaks
The most important implication of frequent structral breaks is that we have difficulty assembling a lengthy data period where a single stable model is relevant. The problem is particularly acute in applying time series data to issues in macroeconomics.
Posted by bparke at 07:32 PM | Comments (0)