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MGT611AC-FA18-Exam1-Summary.pptMGT611AC:
Advanced Quantitative Methods
Fall 2018
Christopher P. Wright
Exam 1 Summary of Main Concepts*
* Note: this is not meant to be an exhaustive list of material that may be covered. Unless explicitly excluded, all material discussed in class is fair game.
Model: assumed actual (population) relationship between the dependent var. (y) and indep.var.’s (x’s).
- β0 = y intercept (average y when all xi = 0.)
- βi = slope of y with respect to xi holding all else constant.
- Example:
Multiple Regression
ɛ = “noise”
Estimated Regression Equation: estimated relationship (based on sample) between the average dependent var. (y) and the indep.var.’s (x’s).
- Example:
- Interpreting the regression coefficients:
- b0 represents an estimate of the value of y when all independent variables (x’s) have a value of 0.
- bi represents an estimate of the change in y corresponding to a 1-unit increase in xi when all other independent variables are held constant.
Multiple Regression
Multiple Coefficient of Determination
- Mult. Coef. of Determ. (R2) – % of variation in y within the sample explained by relationship with x’s:
- Measures fit of model to data (close to 1 is good; to 0 is bad.)
- Adjusted Mult. Coef. of Determination (Ra) – adjusts R2 for number of x’s relative to sample size, n:
- Used to compare models of different sizes, since adding x’s can increase (never decreases) R2 even without real relationships.
The hypotheses are:
Test statistic:
F-Test of Overall Signif.
(Critical Value approach)
H0: there is not a relationship between y and any of our x’s.
HA: there is a relationship between y and at least one of our x’s.
(p-Value approach)
Reject if:
DOF = (p, n – p –1)
Given by software
Reject if:
If reject (2 parts):
- Proven at least one slope (βi) is not 0;
- y has a relationship w/ at least one xi .
Reject
p-value
1 – α
α
1
H0
F
The hypotheses are:
Test statistic:
t-Test of Individual Sign.
(Critical Value approach)
H0: there is not a relationship between y and xi.
HA: there is a relationship between y and xi.
(p-Value approach)
Reject if:
DOF = n – p – 1
Given by software
Reject if:
H0: the average value of y does not depend on xi.
HA: the average value of y does depend on xi.
If reject (2 parts):
- Proven slope of xi (βi) is not 0;
- y has a relationship w/ xi .
t
0
α/2
Reject
p-value/2
tα/2
Reject
tα/2
Multiple Regression Output
Coefficient (bi) Estimates
Coef. of Determ. (R2)
p-value of t-Test
Adj. Coef. of Determ. (Ra2)
Sample Size (n)
p-value of F-test
Dummy Variables – {0, 1} variables used to separate data points into groups (cases)
- For k cases, need k – 1 dummy variables.
- Intercept covers the remaining (all 0’s) “base case”.
- Example:
- Should use generalized F-test for set first (not on exam.)
- If t-test of a case’s slope is signif. (reject), then have signif. evidence of difference between that case and the base case.
Qualitative Variables
Interpreting the regression coefficients of dummy variables:
- bj estimated avg. difference in y between state j and base case (all dummies in set are 0) when all else is held constant.
- b0 represents an estimate of the value of y for base case (all dummies in set are 0) and when all other independent variables have a value of 0.
- Note: (b0 + bj) is the estimate of the intercept for case j
Dummy Variables
Interaction –slope of one x changes with the value of the another x and vice versa.
- Model interaction of xi and xj with new variable:
- If the t-test for slope of xint is significant (reject the Null), then have signif. evidence of an interaction between xi and xj.
- bint is estimated change in slope of xi with a unit change (increase by 1) in xj when all other x’s are held constant, and vice versa.
Interaction
Breusch-Pagan – tests for dependence of error variances on independent variables.
- After running regression, create new variable from residuals:
- Create/run new model w/ dependent variable, e2, and independent variables, x:
- If significant overall (F), then heteroskedasticity is a problem.
- White test adds squares and cross-products of x’s.
Heteroskedasticity
Multicollinearity – two (or more) of the independent variables are too highly correlated.
- For just two x’s, look at correlation:
- If > 0.7 or < -0.7, then have signif. evidence of multicollinearity.
- For two or more x’s, use VIF:
- For each x, run regression of that x (as y) vs. all remaining x’s.
- Read off R2 from each.
- Calculate each VIF:
- If any VIF > 5, then suff. evidence to prove multicol. is a problem.
Multicollinearity
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