1. Returning to Consumption

2. More on Consumption
We return to the consumption problem to illustrate the issue of heteroscedasticity
It turns out that OLS may NOT give us the best estimate of the MPC
The reason is that one of the assumptions of the GM theorem is probably violated in the consumption model
The data is probably not heteroscedastic
Var(ui|xi) ≠ 2

3. Homoscedastic

4. Heteroscedastic

5. Characteristics of Heteroscedasticty
Systematic pattern exists in variance of residuals:
 Var(ui|xi) = i2 = 2.f(xi)
i2 = f(1+2Z2+…+ kZk)
variance has different values for different observations or groups of observations
Intuition: if random bit come from roll of dice then homo is with same dice and hetero is with different dice
Evident in cross-section data or time series

6. Consequences OLS is unbiased OLS is consistent
OLS is no longer efficient
Variance formula used previously is incorrect
significance test, confidence intervals etc. cannot be used
Aside: a corrected formula can be used
Stata: regress y x, robust 
We don’t bother with this because can do better with alternative estimator

7. Testing for Heteroskedasticity
Plot of residuals
Sort the residuals by explanatory variable and plot against that variable , look for pattern, do this for each explanatory variable
Not a formal test but can give an idea of what's going on
Can use it to reject idea of Het

8. An example of Het

9. Consumption Example

10. Goldfeld Quandt test
Used for i2 = 2.f(xi) i.e. related to one variable only
State Hypothesis
Test H0: i2 = 2. H0: i2 ≠ 2.
Note: the null is homoscedasticity
Sort residuals by ascending order of xi
Omit middle 20% observations: (n-c) observations remain
Estimate the original model separately for two samples
first (n-c)/2 obs (keep RSS1 )
last (n-c)/2 obs (keep RSS2)
Compute: g = SSR2/SSR1
If g > Fc(df,df) => reject null hypothesis of homoscedasticity at a significance level
Test can be carried out for each xi

11. Intuition of GQ
If het does exist then we can split sample into a low variance and high variance bit
Run the regression separately for the two samples
Calculate the ratio of variances of the residual (remember s2=RSS/df)
If this ratio is 1 then they are equal and the data is homoscedastic
So reject null of homoscedasticity if bigger than 1
How much bigger? Bigger than F critical value

12. Consumption Example Test it for nmwage State Hypothesis
Test H0: i2 = 2 H0: i2 ≠ 2.
Note: the null is homoscedasticity
Sort residuals by ascending order of nmwagei
Stata command: sort nmwage
Omit middle 20% observations: (n-c) observations remain
Two sample:
Estimate the original model separately
Compute: g = SSR2/SSR1
g= / =
If g > Fc(df,df) => reject null hypothesis of homoscedasticity at a significance level
5% sig level F(550,550)=1.15
So cannot reject the null at 5% significance level
Test can be carried out for each xi

13. sort nmwage . regress cons nmwage if _n<=550 Source | SS df MS Number of obs = F( 1, 548) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = cons | Coef. Std. Err. t P>|t| [95% Conf. Interval] nmwage | _cons | regress cons nmwage if _n> F( 1, 548) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = nmwage | _cons |

14. White’s Test
More general test that allows for more than one varibles to influence the variance of the residuals
Estimate model yi = 1 + 2 x2i + 3 x3i + ui
Run auxiliary regression: sq’d residuals on squares and cross products of X variables:
ei2 =1 +2 x2i +3 x3i + 4 x2i2+ 5 x3i2+ 6 x2i x3i + vi
Null hypothesis is homoscedastic errors i.e.
2 = 3 = 4 = 5 = 6 = 0 i.e. ei2 = constant
calculate nR2 ~ df2 test
nR2 > df2 critical value reject null hypothesis
Comment: why not an F-test

15. Consumption Example Test it for nmwage State Hypothesis
Test H0: i2 = 2 H0: i2 ≠ 2.
Note: the null is homoscedasticity
Estimate the Model and generate residuals squared
Regress residual squared on all of the variables that may cause heteroscdasticity
Form the test statistic: NR2=0.266
Find critical value: chi-sq, df=2 alpha=0.05=5.99
We cannot reject the null at the 5% significance level

16. predict u, residual gen u2=u^2 gen nmwage2=nmwage^2 regress u2 nmwage nmwage2 Source | SS df MS Number of obs = F( 2, 1327) = 0.11 Model | e e+09 Prob > F = Residual | e e+10 R-squared = Adj R-squared = Total | e e+10 Root MSE = 2.3e u2 | Coef. Std. Err. t P>|t| [95% Conf. Interval] nmwage | nmwage2 | _cons |

17. Efficient Estimation
If we find heteroscsadstity we know that OLS will be inefficient
Remember why this might be a problem (see over)
Can we do better?
Yes. There is an efficient estimator called Generalised Least Squares (GLS)
Two steps
Remove the heteroscedasticity from the data
Do OLS on the transformed data

18. Prob of error is lower for efficient estimator at any sample size
Same sample size, different estimator

19. The GLS Procedure Assume that i2 is known: Basic model:
Yi = 1 + 2 Xi + ui , E(ui2) = i2 (not constant)
Create new data with each observations weighted by the heteroscedastic standard deviation
𝑌 𝑖 ∗ = 𝑌 𝑖 𝜎 𝑖 𝑋 2𝑖 ∗ = 𝑋 2𝑖 𝜎 𝑖 𝑋 1𝑖 ∗ = 1 𝜎 𝑖

20. The GLS Procedure Then run the regression on the transformed data
Yi∗ = 1∗ 𝑋 1𝑖 ∗ + 2 X2i∗ + 𝑢 𝑖 ∗
The slope estimates are the BLUE of the coefficients of the original model
Note the intercept tem is slightly different (it has now become coefficient on a variable)

21. How it Works GLS eliminates heteroscedasticity To see this note that
var(ui*) = E(ui*)2 = E(ui/i)2 = 1/i2.E(ui2) = (1/i2).i2 = 1
var of transformed error term is homoskedastic: it is constant
NB This model does not have a constant now: it has two explanatory variables: 1/i and Xi/I
Cannot apply GLS if the exact type of hetero is unknown. So do FGLS (Feasible GLS) and replace i with an estimate of I
From White’s test

22. The Consumption Example
Transform the data to eliminate the heteroscedasticty
Use the estimate of from White’s test
Stata command
Predict white
generate c=cons/sqrt(white)
generate y=nmwage/sqrt(white)
The GLS of the MPC is given by the regression on the transformed data

23. predict white (option xb assumed; fitted values). gen sigma=white^0. 5
predict white (option xb assumed; fitted values) . gen sigma=white^ gen c=cons/sigma . gen y=nmwage/sigma . regress c y Source | SS df MS Number of obs = F( 1, 1328) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = c | Coef. Std. Err. t P>|t| [95% Conf. Interval] y | _cons |

24. Conclusion The example didn’t appear to have heteroscedasticity.
When het does exist the difference between GLS and OLS can be substantial
Both are unbiased and consistent
GLS is preferable because it is efficient so there is a lower probability of substantial error