1. Multivariate Regression

2. Topics The form of the equation Assumptions
Axis of evil (collinearity, heteroscedasticity and autocorrelation)
Model miss-specification
Missing a critical variable
Including irrelevant variable (s)

3. The form of the equation
Yt= Dependent variable
a1 = Intercept
b2= Constant (partial regression coefficient)
b3= Constant (partial regression coefficient)
X2 = Explanatory variable
X3 = Explanatory variable
et = Error term

4. Partial Correlation (slope) Coefficients
B2 measures the change in the mean value of Y per unit change in X2, while holding the value of X3 constant. (Known in calculus as a partial derivative)
Y = a +bX
dy = b

5. Assumptions of MVR
X2 and X3 are non-stochastic, that is, their values are fixed in repeated sampling
The error term e has a zero mean value (Σe/N=0)
Homoscedasticity, that is the variance of “e”, is constant.
No autocorrelation exists between the error term and the explanatory variable.
No exact collinearity exist between X2 and X3
The error term “e” follows the normal distribution with mean zero and constant variance

6. Venn Diagram: Correlation & Coefficients of Determination (R2)Y Y X2 X1 X1 X2
Correlation exists between
X1 and X2. There is a portion of the
variation of Y that can be attributed to
either one
No correlation exists between
X1 and X2. Each variable explains
a portion of the variation of Y

7. A special case: Perfect Collinearity
X1 X2
X2 is a perfect function of X1. Therefore, including X2 would be irrelevant because does not explain any of the variation on Y that is already accounted by X1. The model will not run.

8. Consequences of Collinearity
Multicollinearity is related to sample-specific issues
Large variance and standard error of OLS estimators
Wider confidence intervals
Insignificant t ratios
A high R2 but few significant t ratios
OLS estimators and their standard error are very sensitive to small changes in the data; they tend to be unstable
Wrong signs of regression coefficients
Difficult to determine the contribution of explanatory variables to the R2

9. TESTING FOR MULTICOLLINARITY

10. DEPENDENT
TLA
BATHS
BEDROOM
AGE

11. IS BAD IF WE HAVE MULTICOLLINEARITY?
If the goal of the study is to use the model to predict or forecast the future mean value of the dependent variable, collinearity may not be a problem
If the goal of the study is not prediction but reliable estimation of the parameters then collinearity is a serious problem
Solutions: Dropping variables, acquire more data or a new sample, rethinking the model or transform the form of the variables.

12. Heteroscedasticity
Heteroscedasticity: The variance of “e” is not constant, therefore, violates the assumption of hemoscedasticity or equal variance.

13. Heteroscedasticity

14. What to do when the pattern is not clear ?
Run a regression where you regress the residuals or error term on Y.

15. LET’S ESTIMATE HETEROSCEDASTICITY
Do a regression where the residuals become the dependent
Variable and home value the independent variable.

16. Consequences of Heteroscedasticity
OLS estimators are still linear
OLS estimators are still unbiased
But they no longer have minimum variance. They are not longer BLUE
Therefore we run the risk of drawing wrong conclusions when doing hypothesis testing (Ho:b=0)
Solutions: variable transformation, develop a new model that takes into account no linearity (logarithmic function).

17. Testing for Heteroscedasticity
Let’s regress the predicted value (Y hat) on the log of the residual
(log e2) to see the pattern of heteroscedasticity.
Log e2
The above pattern shows that our relationships is best described as a
Logarithmic function

18. Autocorrelation
Time-series correlation: The best predictor of sales for the present Christmas season is the previous Christmas season
Spatial correlation: The best predictor of a home’s value is the value of a home next door or in the same area or neighborhood.
The best predictor for a politician, to win an election as an incumbent, is the previous election (ceteris paribus)

19. The product of two different error terms Ui and Uj is zero.
Autocorrelation
Gujarati defines autocorrelation as “correlation between members of observations ordered in time [as time- series data] or space as [in cross-sectional data].
E (UiUj)=0
The product of two different error terms Ui and Uj is zero.
Autocorrelation is a model specification error or the regression model is not specified correctly. A variable is missing or has the wrong functional form.

20. Types of Autocorrelation

21. The Durbin Watson Test (d) of Autocorrelation
Values of the d
d = 4 (perfect negative correlation
d = 2 (no autocorrelation)
d = 0 (perfect positive correlation)

22. Let’s do a “d” test
Here we solved the problem of collinearity, heteroscedasticity and
autocorrelation. It cannot get any better than this.

23. Model Miss-specification
Omitted variable bias or underfitting a model. Therefore
The omitted variable is correlated with the included variable then the parameters estimated are bias, that is their expected values do not match the true value
The error variance estimated is bias
The confidence intervals and hypothesis-testing procedures and unreliable.
The R2 is also unreliable
Let’s run a model
LnVAL = a + bLNTLA + bLNBDR + bLNAGE (true model)
LnVAL=a +bLNBDR + LNAGE + e (underfitted)

24. Model Miss-specification
Irrelevant variable bias
The unnecessary variables has not effect on Y (although R2 may increase).
The model still give us unbias and consistent estimates of the coefficients
The major penalty is that the true parameters are less precise therefore the CI are wider increasing the risk of drawing invalid inference during hypothesis testing (accept the Ho: B=0)
Let’s run the following model:
LNVALUE=a + bLNTLA+ bLNBTH + bLNBDR + bLNAGE