3 3 Omitted Variable Bias When a valid

3. 3 Omitted Variable Bias -When a valid variable is excluded, we UNDERSPECIFY THE MODEL and OLS estimates are biased -Consider the true population model: -Assume this satisfies all 4 assumptions and that we are concerned with x 1 -if we exclude x 2, our estimation becomes:

3. 3 Omitted Variable Bias -From (3. 23) we know that: -where Bhats come from regressing y on ALL x’s and deltatilde comes from regressing x 2 on x 1 -since deltatilde depends on independent variables, it is considered fixed -we also know from Theorem 3. 1 that Bhats are unbiased estimators, therefore:

3. 3 Omitted Variable Bias -From this we can calculate Btilde’s bias: -this bias is often called OMITTED VARIABLE BIAS -From this equation, B 1 tilde is unbiased in two cases: 1) B 2=0; x 2 has no impact on y in the true model 2) deltatilde=0

3. 3 Deltatilde=0 -deltatilde is equal to the covariance of x 1 and x 2 over the variance of x 1, all in the sample -deltatilde is equal to zero only if x 1 and x 2 are uncorrelated -therefore if they are uncorrelated, B 1 hat is unbiased -it is also unbiased if we can show that:

3. 3 Omitted Variable Bias -As B 1 hat’s bias depends on B 2 and deltatilde, the following table summarizes the possible biases: Corr(x 1, x 2)> Corr(x 1, x 2)< 0 0 B 2 hat>0 Positive Bias Negative Bias B 2 hat<0 Negative Bias Positive Bias

3. 3 Omitted Variable Bias -the SIZE of the bias is also important, as a small bias may not be cause for concern -therefore the SIZE of B 2 and deltatilde are important -although B 2 is unknown, theory can give us a good idea about its sign -likewise, the direction of correlation between x 1 and x 2 can be guessed through theory -a positive (negative) bias indicates that given random sampling, on average your estimates will be too large (small)

3. 3 Example Take the true regression: Where pasta taste depends on experience making pasta and love -While we can measure years of experience, we can’t measure love, so we find that: What is the bias?

3. 3 Example We know that the true B 2 should be positive; love improves cooking We can also support a positive correlation between experience and love, if you love someone you spend time cooking for them Therefore B 1 hat will have a positive bias However, since the correlation between experience and love is small, the bias will likewise be small

3. 3 Bias Notes -It is important to realize that the direction of bias is ON AVERAGE -a positive bias on average may underestimate in a given sample If There is an UPWARD BIAS If There is a DOWNWARD BIAS And B 1 tilde is BIASED TOWARDS ZERO if it is closer to zero than B 1

3. 3 General Omitted Bias Deriving the direction of omitted variable bias with more independent variables is more difficult -Note that correlation between any explanatory variable and the error causes ALL OLS estimates to be biased. -Consider the true and estimated models: x 3 is omitted and correlated with x 1 but not x 2 Both B 1 tilde and B 2 tilde will always be biased unless x and x are uncorrelated

3. 3 General Omitted Bias Since our x values can be pairwise correlated, it is hard to derive the bias for our OLS estimates -If we assume that x 1 and x 2 are uncorrelated, we can analyze B 1 tilde’s bias without x 2 having an effect, similar to our 2 variable regression: With this formula similar to (3. 45), the previous table can be used to determine bias -Note that much uncorrelation is needed to determine bias

3. 4 The Variance of OLS Estimators -We now know the expected value, or central tendency, of the OLS estimators -Next we need information on how much spread OLS has in its sampling distribution -To calculate variance, we impose a HOMOSKEDASTICITY (constant error variance) assumption in order to 1) Simplify variance formulas 2) Give OLS an important efficiency property

Assumption MLR. 5 (Homoskedasticity) The error u has the same variance given any values of the explanatory variables. In other words,

Assumption MLR. 5 Notes -MLR. 5 assumes that the variance of the error term, u, is the SAME for ANY combination of explanatory variables -If ANY explanatory variable affects the error’s variance, HETEROSKEDASTICITY is present -The above five assumptions are called the GAUSS-MARKOV ASSUMPTIONS -As listed above, they apply only to crosssectional data with random sampling -time series and panel data analysis require more complicated, related assumptions

Assumption MLR. 5 Notes If we let X represent all x variables, combining assumptions 1 through 4 give us: Or as an example: MLR. 5 can be simplified to: Or for example:

3. 4 MLR. 4 vs. MLR. 5 “Assumption MRL. 4 says that the expected value of y, given X, is linear in the parameters – but it certainly depends on x 1, x 2, …. , xk. ” “Assumption MLR. 5 says that the variance of y, given X, does not depend on the values of the independent variables. ” (bold added)

Theorem 3. 2 (Sampling Variances of the OLS Slope Estimators) Under assumptions MLR. 1 through MRL. 5, conditional on the sample values of the independent variables, For j= 1, 2, …, k, where Rj 2 is the R-squared from regressing xj on all other independent variables (and including an intercept) and SST is the total sample variation in xj:

Theorem 3. 2 Notes Note that all FIVE Gauss-Markov assumptions were needed for this theorem Homoskedasticity (MLR. 5) wasn’t needed to prove OLS bias The size of Var(Bjhat) is very important -a large variance leads to larger confidence intervals and less accurate hypothesis tests