Lecturer: Ing. Martina Hanová, PhD.
How do we evaluate a model? How do we know if the model we are using is good? assumptions relate to the (population) prediction errors study through the (sample) estimated errors, the residuals
The four conditions of the SRL model What can be wrong with our model? 1. Yi is a Linear function of the Xi 2. Errors ε i, are Independent. 3. Errors ε i are Normally distributed. 4. Errors ε i have Equal variances (denoted σ 2 ).
all of the estimates, intervals, and hypothesis tests arising in a regression analysis have been developed assuming that the model is correct. all the formulas depend on the model being correct! if the model is incorrect, then the formulas and methods (OLS) we use are at risk of being incorrect.
Model is linear in parameters
LRM NRM
Zero mean value of disturbances - ui.
No autocorrelation between the disturbances The data are a random sample of the population
Equal variance of disturbences - ui homoskedasticity Errors have constant variance “homoskedasticity” heteroskedasticity Errors have non-constant variance “heteroskedasticity”
Construction of var-cov matrix: vector ei * transpose vector ei
The errors are normally distributed Normal Probability Plot
The errors, i, are independent normal random variables independent normal random variables with mean zero and constant variance, σ2
The population regression function is not linear. The error terms are not independent. The error terms are not normally distributed. The error terms do not have equal variance.
Zero covariance between ui and Xi
the number of >= the number of observations explanatory variables
The mean of the response, E(Yi), at each set of values of the predictor, (x1i,x2i,…), is a Linear function of the predictors. The errors, ε i, are Independent. The errors, ε i, at each set of values of the predictor, (x1i,x2i,…), are Normally distributed. The errors, ε i, at each set of values of the predictor, (x1i,x2i,…) have Equal variances (denoted σ 2 ).
All tests and intervals are very sensitive to even minor variance from independence. All tests and intervals are sensitive to moderate variance from equal variance. The hypothesis tests and confidence intervals for β i are fairly "robust" (that is, forgiving) against variance from normality. Prediction intervals are quite sensitive to variance from normality.
The Gauss Markov theorem When the first 4 assumptions of the simple regression model are satisfied the parameter estimates are unbiased and have the smallest variance among other linear unbiased estimators. The OLS estimators are therefore called BLUE for Best Linear Unbiased Estimators