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### Economics 105: Statistics

### Multiple Regression

### Multiple Regression

### Multiple Regression

### Specification Bias

### Specification Bias

### Omitted Variable Bias

### Omitted Variable Bias

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

I know! We can save the model, but not until Eco205.

Holy endogeneity, Batman!

Violations of GM AssumptionsAssumption Violation

Wrong functional form

Omit Relevant Variable (Include Irrelevant Var)

Errors in Variables

Sample selection bias, Simultaneity bias

“well-specified model” (1) & (5)

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

can only assess theoretically

zero conditional mean of errors (2)

Homoskedastic errors (3)

Heteroskedastic errors

No serial correlation in errors (4)

There exists serial correlation in errors

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

Assumptions

(3)

Homoskedasticity

(4)

No autocorrelation

(5)

Errors and the explanatory variables are uncorrelated

(6)

Errors are i.i.d. normal

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

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

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

When is an unbiased estimator of ?

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

… approximately equal

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