Ordinary Least Squares Ordinary Least Squares – a regression estimation technique that calculates the Beta-hats -- estimated parameters or coefficients of the model – so as to minimize the sum of the squared residuals. Σei2 = Σ (Yi – Yhat)2 Where e = ε = the residual Yhat is the predicted value of Y

Why OLS? OLS is relative easy to use. For a model with one or two independent variables you one can run OLS with a simple spreadsheet (without using the regression function). The goal of minimizing the sum of squared residuals is appropriate from a theoretical point of view. OLS estimates have a number of useful characteristics.

Why not minimize residuals? Residuals can be positive and negative. Just minimizing residuals can produce an estimator with large positive and negative errors that cancel each other out. Minimizing the absolute value of the sum of residuals poses mathematical problems. Plus, we wish to minimize the possibility of very large errors.

Useful Characteristics of OLS Estimated regression line goes through the means of Y and X. In equation form Mean of Y = Β0 + Β1(Mean of X) The sum of residuals is exactly zero. OLS estimators, under a certain set of assumptions (which we discuss later), are BLUE (Best Linear Unbiased Estimators). Note: OLS is the estimator, the coefficients or parameters are the estimates.

Classical Assumption 1 The error term has a zero population mean. We impose this assumption via the constant term. The constant term equals the fixed portion of the Y that cannot be explained by the independent variables. The error term equals the stochastic portion of the unexplained value of Y.

Classical Assumption 2 The error term has a constant variance Heteroskedasticity Where does this most often occur? Cross- sectional data Why does this occur in cross-sectional data?

Classical Assumption 3 Observations of the error term are uncorrelated with each other. Serial Correlation or Autocorrelation Where does this most often occur? Time- series data Why does this occur in time-series data?

Classical Assumption 4-5 The data for the dependent variable and independent variable(s) do not have significant measurement errors. The regression model is linear in the coefficients, is correctly specified, and has an additive error term.

Classical Assumption 6 The error term is normally distributed This is an optional assumption, but a good idea. Why? One cannot use the t-statistic or F-statistic unless this holds (will explain these later).

Five More Assumptions All explanatory variables are uncorrelated with the error term. When would this not be the case? Then a system of equations is needed (i.e. supply and demand). What are the consequences? Estimation of slope coefficient for correlated X terms is biased. No explanatory variable is a perfect linear function of any other explanatory variable. Perfect collinearity or multicollinearity Consequence: OLS cannot distinguish the impact of each X on Y. X values are fixed in repeated sampling. The number of observations n must be greater than the number of parameters to be estimated. There must be variability in X and Y values.

The Gauss-Markov Theorem Given Classical Assumptions, the OLS estimator βk is the minimum variance estimator from among the set of all linear unbiased estimators of βk. In other words, OLS is BLUE Best Linear Unbiased Estimator Where Best = Minimum Variance

Given assumptions… The OLS coefficient estimators will be unbiased have minimum variance are consistent are normally distributed. The last characteristic is important if we wish to conduct statistical tests of these estimators, the topic of the next chapter.

Unbiased Estimator and Small Variance Unbiased estimator – an estimator whose sampling distributions has as its expected value the true value of β. In other words... the mean value of the distribution of estimates equals the true mean of the item being estimated. In addition to an unbiased estimate, we also prefer a “small” variance.

How does OLS Work? The Univariate Model Y = β0 + β1X + ε For Example: Wins = β 0 + β1Payroll + ε How do we calculate β1? Intuition: β1 equals the joint variation of X and Y (around their means) divided by the variation of X around its mean. Thus it measures the portion of the variation in Y that is associated with variation in X. In other words, the formula for the slope is: Slope = COV(X,Y)/V(X) or the covariance of the two variables divided by the variance of X.

How do we calculate β1? Some Simple Math β1 = Σ [(Xi - mean of X) * (Yi - mean of Y)] / Σ (Xi - mean of X)2 If xi = Xi - mean of X and yi = Yi - mean of Y then β1 = Σ[(xi*yi)] / Σ(xi)2

How do we calculate β0? Some Simple Math β0 = mean of Y – β1*mean of X β0 is defined to ensure that the regression equation does indeed pass through the means of Y and X.

Multivariate Regression Multivariate regression – an equation with more than one independent variable. Multivariate regression is necessary if one wishes to impose “ceteris paribus.” Specifically, a multivariate regression coefficient indicates the change in the dependent variable associated with a one-unit increase in the independent variable in question holding constant the other independent variables in the equation.

Omitted Variables, again If you do not include a variable in your model, then your coefficient estimate is not calculated with the omitted variable held constant. In other words, if the variable is not in the model it was not held constant. Then again.... there is the Principle of Parsimony or Occam’s razor (that descriptions be kept as simple as possible until proven inadequate). So we don’t typically estimate regressions with “hundreds” of independent variables.

The Multivariate Model Y = β0 + β1X1 + β2X2 + ....... βnXn + ε For Example, a model where n=2: Wins = β 0 + β1PTS + β2DPTS + ε Where PTS = Points scored in a season DPTS = Points surrendered in a season

How do we calculate β1 and β2? Some Less Simple Math Remember: xi = Xi - mean of X and yi = Yi - mean of Y β1 = [Σ(x1*yi)*Σ(x2)2 - Σ(x2*yi)*Σ(x1*x2) ] / [Σ(x1)2 * Σ(x2)2 - Σ(x1*x2 )2] β2 = [Σ(x2*yi)*Σ(x1)2 - Σ(x1*yi)*Σ(x1*x2 )] / β0 = mean of Y – β1*mean of X – β2*mean of X

Issues to Consider when reviewing regression results Is the equation supported by sound theory? How well does the estimated regression fit the data? Is the data set reasonably large and accurate? Is OLS the best estimator for this equation? How well do the estimated coefficients correspond to our expectations? Are all the obviously important variables included in the equation? Has the correct functional form been used? Does the regression appear to be free of major econometric problems? NOTE: This is just a sample of questions one can ask.