1Assumptions of Ordinary Least Squares Regression(Part 1)ESM 206Jan 17, 2008

2Assumptions of OLS regression1.Model is linear in parameters2.The data are a random sampleof the population1.The errors are statisticallyindependentfrom one another3.The expected value of the errors is always zero4.The independent variables are not too strongly collinear5.The independent variables are measured precisely6.The residuals have constant variance7.The errors are normally distributed•If assumptions 1-5 are satisfied, thenOLS estimator is unbiased•If assumption 6 is also satisfied, thenOLS estimator has minimum varianceof all unbiased estimators.•If assumption 7 is also satisfied, then we can do hypothesis testing using tand Ftests•How can we test these assumptions?•If assumptions are violated,–what does this do to our conclusions?–how do we fix the problem?

31. Model not linear in parameters•Problem:Can’t fit the model!•Diagnosis:Look at the model•Solutions:1.Re-frame the model2.Use nonlinear least squares (NLS) regression

42. Errors not independent•Problem:parameter estimates are biased•Diagnosis (1):look for correlation between residuals and another variable (not in the model)–I.e., residuals are dominated by another variable, Z, which is not random with respect to the other independent variables•Solution (1):add the variable to the model•Diagnosis (2):look at autocorrelation functionof residuals to find patterns in –time–Space–I.e., observations that are nearby in time or space have residuals that are more similar than average•Solution (2):fit model using generalized least squares (GLS)