Economics 105: Statistics• Go over GH 24

Risks in Model Building• Including irrelevant X’s

– Increases complexity– Reduces adjusted R2

– Increases model variability across samples• Omitting relevant X’s

– Fails to capture fit– Can bias other estimated coefficients

• Where omitted X is related to both other X’s and to the dependent variable (Y)

More Risks:Samples Can Mislead

• Remember: we are using sample data– About 5% of the time, our sample will include

random observations of X’s that result in betahat’s that meet classical hypothesis tests

– Or the beta’s may be important, but the sample data will randomly include observations of X that do not meet the statistical tests

• That’s why we rely on theory, prior hypotheses, and replication

Violations of GM AssumptionsAssumption Violation

“well-specified model” (1) &

(5)

zero conditional mean of errors (2)

Wrong functional formOmit Relevant Variable (Include Irrelevant Var)Errors in VariablesSample selection bias, Simultaneity bias

No serial correlation in errors (4)

constant, nonzero mean due to systematically +/- measurement error in Y

can only assess theoretically

Heteroskedastic errors

Homoskedastic errors (3)

There exists serial correlation in errors

I know! We can save the model,

but not until Eco205.

Holy endogeneity,

Batman!

Multiple Regression•Assumptions• (1)

– Linear function in the parameters, plus error– Variation in Y is caused by , the error (as well as X)

• (2) – Sources of error

• Idiosyncratic, “white noise” • Measurement error on Y• Omitted relevant explanatory variables

– If (2) holds, we have exogenous explanatory vars– If some Xj is correlated with error term for some reason, then that Xj is an endogenous explanatory var

Multiple Regression•Assumptions• (3)

– Homoskedasticity•(4)

– No autocorrelation• (5)

– Errors and the explanatory variables are uncorrelated

• (6)– Errors are i.i.d. normal

Multiple Regression• Assumption (7) No perfect multicollinearity

– no explanatory variable is an exact linear function of other X’s– Venn diagram

• Other implicit assumptions – data are a random sample of n observations from proper population– n > K, and ideally n much greater than K– the little xij’s are fixed numbers (the same in repeated samples) or they are realizations of random variables, Xij, that are independent of error term & then inference is done CONDITIONAL on observed values of xij’s

Specification Bias• Violation of Assumptions (1 & 5): well-specified model• true model is (A)

but we run (B)

• Including an irrelevant variable• is an unbiased estimator of

• ; less efficient

• estimator of , , is unbiased• t & F tests are valid

Specification Bias• Violation of Assumptions (1&5): well-specified model• true model is (C)

but we run (D)

• Omitting a relevant variable• is a biased estimator of

• is actually smaller; more efficient

• estimator of , , is now biased• t & F tests are incorrect

Omitted Variable Bias•

• When is an unbiased estimator of ?

• b21 is the slope coefficient from a regression of the EXCLUDED variable on the INCLUDED variable

Omitted Variable Bias• • Subcript c indexes 64 countries •Descriptive statistics

Omitted Variable Bias• •

Omitted Variable Bias• •

Omitted Variable Bias• •

Omitted Variable Bias• •

Omitted Variable Bias• •

• • … approximately equal