Least squares linear regression continued

If Yi equals 0, the model becomes partially recursive. The first equation becomes a regression which can be estimated by ordinary least squares. However, the second equation continues to fail the order condition. To see the problem, consider that even with the restriction, any linear combination of the two equations has the same variables as the original second eqation. 

Suppose that a linear probability model is to be fit to a set of observations on a dependent variable, y, which takes values zero and one, and a single regressor, x, which varies continuously across observations. Obtain the exact expressions for the least squares slope in the regression in terns of the mean(s) and variance of x and interpret the result. 

The methods of data analysis depend on the nature of the final output. If the problem is one of classification, a number of multivariate classifiers are available such as those based on principal components analysis (SIMCA), cluster analysis and discriminant analysis, or non-linear artificial neural networks. If the required output is a continuous variable, such as a concentration, then partial least squares regression or principal component regression are often used [20]. 

McCulloch (1971) proposes a more practical approach, using polynomial splines. This method produces a fimction that is both continuous and linear, so the ordinary least squares regression technique can be employed. A 1981 study by James Langetieg and Wilson Smoot, cited in Vasicek and Fong (1982), describes an extended McCulloch method that fits cubic splines to zero-coupon rates instead of the discount fimction and uses nonlinear methods of estimation. 

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