December 08, 2005
Logit/Probit
If the dependent variable is binary (two possible values) or categorical, we need a new class of models. The strategy is to find a statistical model that accounts for a discrete dependent variable. Estimating that model then yields parameter estimates and standard errors, which we can analyze using the same techniques we studied for regression models.
The probit model is based on the normal probability density and cdf. It has an elegant mathematical basis, but presented challenges to early computational capabilities.
The logit model has very similar properties, but had the early advantage of being easily differentiable.
There are more versions of these models to explain more complicated choices.
Posted by bparke at 09:32 PM | Comments (0)
December 06, 2005
Simultaneity
If the regressors are not exogenous, we need to worry about simultaneity bias. That is, the parameters estimates are inconsistent, which means that they are biased and the bias does not go away in large samples.
A basic supply and demand model nicely illustrates the main concepts. There are two curves in the model, and a regression tries to fit one line to the data.
Details:
Here is how both curves shifting generates the data. Shifts in either curves change both price and quantity.
Exclusion restrictions that cause a given variable to shift just one curve help to identify the other curve.
Indirect least squares shows how we can recover estimates of the structural parameters from the reduced form parameters if exogenous variables are excluded from some equations.
If an equation is identified, we can calculate consistent parameter estimates using two-stage least squares.
Posted by bparke at 11:20 PM | Comments (0)
December 01, 2005
Heteroskedasticity
The classic example of heteroskedasticity of a know form and the classic correction:
Avoiding obvious heteroskedasticity is often a matter of using a sensible functional form.
Using logarithms is also a common way to avoid obvious heteroskedasticity.
The ARCH model, which envisions serial correlation in the error variance, has been very popular in explaining volatility clustering in the financial markets.
Posted by bparke at 11:06 PM | Comments (0)