Straight Line Regression:
Y(i) = Beta(0) + Beta(1)*X(i) + Epsilon(i), where Beta(0) is the intercept (constant) and Beta(1) is the slope (coefficient) and Epsilon(i) is white noise
iid w/ mean 0 and variance (sigma)^2.
Least Squares Estimation:
sum{Y(i) – (b(0) + b(1)*X(i))}^2, where Y(i) is the observed value and (b(0) + b(1)*X(i)) is from the model.
We have a null hypothesis that says that the coefficients (beta(i)’s) are zero.
If we have a low pvalue/high Fvalue, we reject this hypothesis, and can say that the coefficients are
not 0, and therefore the variables have a linear relationship to with each other (colinearity).
However, we cannot always trust the low pvalue/high Fvalue because we are testing
multiple variables.
R^2 = 68.5% means that the model explains 68.5% of the variation.
We want a high value for (R^2), but we should be suspicious about an (R^2) value being
too high.
Standard error = sigma.
We want a low standard error.
The degrees of freedom for regression is 0 = number of predictor variables (p=1 for straight line regression).
The total degrees
of freedom is (n1).
The residual error dof is (np1).
The mean sum of squares (MS) is its sum of squares divided by its degrees of freedom.
F = (regression MS)/(residual error
MS).
Least squares line is a sample version of best linear predictor
R^2 is the squared correlation between X and Y is the fraction of the variation that can be predicted using the linear predictor.
Multiple Linear Regression:
Y(i) = Beta(0) + Beta(1)*X(i1) + … + Beta(p)*X(ip) + Epsilon(i).
All coefficients estimated by least squares.
Model Selection:
larger models have less bias (good) and would give us best predictions if all coefficients could be estimated w/o error, but larger models (more coefficients),
when coefficients are replaced by estimates, the prediction becomes less accurate and more variability (bad).
R^2 is not useful for comparing models of different sizes because it will always choose the largest model., so use adjusted R^2.
C(p) estimates how well a model will predict.
We
want to choose where C(p) <= p, but approximately equal.
Nonlinear Regression:
Use this because we can’t use a linear model for nonlinear functions. Need to estimate a starting
r (usually between 5 – 7%), look at SAS output and find the r where the sum of squares is the smallest.
Residual Plotting:
problems to look for: (1) nonnormality – outliers can
be a problem since they have a large influence on the estimation results; solution – transformation of the response, (2) nonconstant variance – causes inefficient (too variable)
estimates; solution – transformation of the response and weighting, (3) nonlinearity, (4) model misspecification means E(YX(1)….X(p)) has a functional form different from the
model; causes biased estimates; solution – transformation, polynomial regression, nonparametric regression, etc.
Ratings:
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 Spring '07
 ANDERSON
 Regression Analysis, Market Portfolio

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