1. Incorrect specification of form of a relationship (eg – non-linear)

2. Omission of variable (misspecification)

3. Measurement error in the dependent variable

4. Variables move smoothly over time (sluggishness – lag e.g. GDP)

In 1 & 2 included IVs the error term is picking up the missing variables

3&4 are true autocorrelation

• 0-4,
• 0=positive autocorrelation,
• 4=perfect negative autocorrelation.
• 2=no autocorrelation

• Carry out auxiliary regression in which the squared residuals (ui) from the main regression are regressed on all of the IVs
• υ_i^2=α_1+α_2 X_2i+α_3 X_3i+α_4 X_2i^2+〖α_5 X_3i^2+α〗_6 X_2i X_3i+ν_i
• Obtain raw R2
• Df = number of regressors (not u) so 5 in above
• N* R2~χ2df
• If calculated chi-squared > critical chi-squared reject the null hypothesis of homoscedasticity

Durbin Watson hypothesis test for 1st order autocorrelation

• If d=2 – no autocorrelation

• If 0≤d≤2 – positive correlation

• If 2≤d≤4 then negative autocorrelation

d<dl - reject H0

d>du - do not reject

dl<d<du - indeterminate range

1. Run OLS on original equation

2. Regress residuals on lagged residuals to estimate rho. 3. Transform variables using this estimate of rho.

4. Run OLS on the transformed variables.

Cochrane-Orcutt Iterative Process - uses GLS approximations in an iterative process to converge on a reliable estimate of ρ for first-order serially correlated disturbances; based on minimizing the sum of squared residuals (RSS) for different values of ρ until a stable value for the RSS is obtained (i.e., one that does not change between iterations)

Origins of simultaneous equation bias

When the dependent variable from one equation becomes the independent variable in another there is a violation of the assumption that the IV has no relation to the error term. This results in inconsistent estimates and bias.

To measure goodness of fit of a logit model. 60 - fair, 70 - good, 80 - vg, 90-excellent

Interpretation of the parameter estimate from a Logit regression.

Beta is the change in the log of the odds of y occurring given a one unit change in x holding the effects of other variables constant

•  The odds of y occurring is the odds ratio multiplied by (parameter estimate) given a one unit increase in the independent variable holding the effects of the other variables contstant.
• Or the multiplicative increase in the odds of the dependent event occurring if the event represented by the independent variable occurs, holding the effects of the other variables contstant.

1. The probability of the event (P_i) is between 0 and 1 ( or 0% to 100%)

2. S shaped curve – so as the value of the IV approaches zero, P_i decreases at a decreasing rate and is asymptotic to 0.

3. Conversely, as the IV get large,P_iincreases at a decreasing rate and is asymptotic to 1.

Decision rule – range because d depends not only on p but also values of the x variables.

·       H₀: p=0

·       If d<d_L – reject

·       If d > d_u  - do not reject

·       If d_L < d < d_u   - cannot confirm or deny

Beta is the change in the log of the odds of y occurring given a one unit change in x holding the effects of other variables constant

The odds of y occurring is the odds ratio multiplied by (parameter estimate) given a one unit increase in the independent variable holding the effects of the other variables contstant. Or the multiplicative increase in the odds of the dependent event occurring if the event represented by the independent variable occurs, holding the effects….

1. The probability of the event (P_i) is between 0 and 1 ( or 0% to 100%) and 2. S shaped curve – so as the value of the IV approaches zero, P_i decreases at a decreasing rate and is asymptotic to 0. Conversely, as the IV get large,P_i increases at a decreasing rate and is asymptotic to 1.

Process:

1.      Regress the endogenous variables (right side) on all the predetermined variables of the model

2.      Replace them with their predicted values

3.      Perform OLS to the transformed version of the equation

1.Beta-hat is biased if there is any correlation with any other included IV.

2.If at all correlated, its effects are picked up by its friends.

3.If at all correlated the variances for the other betas will be smaller than that of the true model

4.The stronger the relationship btw the omitted variable and another IV, the larger the bias.

5.The estimate of the variance of beta-hat is always biased upwards. –what does this mean?

6.Still best

7.F and t-stats are no longer accurate

1.Does not cause bias.

2.Also inflates the standard error of the estimates.

3.OLS is still BLUE.

4.We get an unbiased estimate of a generally inflated variance.

5.F and t-stats are accurate

1.Beta-hat is biased if there is any correlation with any other included IV.

2.If at all correlated, its effects are picked up by its friends.

3.If at all correlated the variances for the other betas will be smaller than that of the true model

4.The stronger the relationship btw the omitted variable and another IV, the larger the bias.

5.The estimate of the variance of beta-hat is always biased upwards. – what does this mean?

6.Still best

7.F and t-stats are no longer accurate

They are correlated. Look at STB, R-sq, and p. It could be that it is not as critical in the model. I would want to do a third regression and drop the now insignificant variable and watch what happens to the R-sq and parameter estimates.

·  Likelihood ratio – overall significance ~F. rejection shows a sig. relation

·  Pseudo R-squared – interpreted the same. Cannot be used between models

·  Parameter estimates – change in the log of the odds of y occurring given a one unit chand in x holding the effects of all other IVs constant

·  Chi-square/ chi-square p-values – just like t-stat and p. report standard error

·  Interpretation of odds ratios – the odds that Y is occurring is multiplied by (odds ratio) given a one unit change in X holding the effect….

·  Concordant – 60% fair, 70% good, 80% very good, 90% exc

o   Pair 1s and 0s

·  AIC and SC – across models – lower is better

o   Predicted probability. Multiply parameter estimate by given values. Complete the model calculation. The result becomes the power to which e is raised. E^x/(1+ E^x)

– overall significance ~F. rejection shows a sig. relation

 interpreted the same as OLS R-squared. Cannot be used between Logit models

– change in the log of the odds of y occurring given a one unit chand in x holding the effects of all other IVs constant

Just like t-stat and p in an OLS regression. Report standard error

– the odds that Y is occurring is multiplied by (odds ratio) given a one unit change in X holding the effect….

– 60% fair, 70% good, 80% very good, 90% exc

oPair 1s and 0s

·Shortcomings – dependent variable is bivariate. Not used for forecasting, need more data to achieve meaningful stable results – at least greater than 100. Needs to use WLS.Interpretation is not as intuitive

Compares significance of the logit regression across models – lower is better

Multiply parameter estimate by given values. Complete the model calculation. The result becomes the power to which e is raised.

e^x/(1+ e^x)

1. sort data by IV in which HTS is suspected
2. Divide data into two datasets (high and low)
3. Remove center observations    (n/5 < c < n/4)
4. RSS-high/RSS-low
5. Calculate df (n-c-2k)/2
6. Compare to critical F
7. null hypothesis is homoscedasticity
8. Transform variables - run again and check for HTS again

Take SAS output and use the parameter estimate of the transformed variable as the intercept and the entercept as the Beta for the transformed variable

1.     Error learning

2.     Increased choice

3.     Data measurement error

4.     Outliers

5.     Correct specification

6.     Skewness

7.     Incorrect form

1.     Still unbiased

2.     Not best – no longer minimum variance

3.     Not efficient

4.     Affects t and F

5.     Affects parameter estimates