Estimation Flashcards
What is y?
Dependent variable
What is Bo?
Intercept/constant coefficient
What are B1-Bk?
Slope coefficients
What are X1-Xk?
Explanatory/independent variables
What is u?
Error term
What is the definition of the causal effect of x on y (under notion of ceteris paribus)?
How y changes if variable x changes, when all other relevant factors are held constant
What does linearity imply about effect of x on y?
Implies a one unit change in x always the same effect on y
- y increases by same value with each 1 unit change in x
What assumptions must hold to ensure the estimators are unbiased in the simple regression model?
- model is linear in parameters
- data is from a random sample
- there must be some variation in our explanatory variable
- no perfect Collinearity between explanatory variables
- there should be no statistical relationship between the error term and regressor (Zero conditional mean assumption)
What is E(U)?
Zero
What is the zero conditional mean assumption?
E(U/X1,X2,……,Xk = 0
- explanatory variables must not contain info about the mean of the unobserved factors
What is the aim of the line/plane of best fit?
We want the residuals to be as small as possible
What does the estimated error term equal?
Estimated u = true value of y - estimated y
What are the algebraic properties of OLS?
- Sum of residuals = 0
- sample covariance between regressors and OLS residuals is zero
- sample averages of y and regressors lie on regression line
What is SST?
Total sum of squares
- represents total variation in dependent variable
What is SSE?
Explained sum of squares
- represents variation explained by regression
What is SSR?
Residual sum of squares
- represents variation not explained by regression
What is SST equal to?
SST = SSE + SSR
What is R-squared?
- measures the proportion of the sample variation in y that is explained by the regression model/explanatory variables
What does R-squared equal?
R-squared = SSE/SST = 1 - SSR/SST
What is the definition of an unbiased estimator?
The expected value of the estimated coefficient = true population value
What assumptions must hold on MLR model so that OLS estimators are unbiased?
MLR.1- Linear in parameters
MLR.2- Random Sampling
MLR.3- No Perfect Collinearity
MLR.4-Zero conditional mean
Describe MLR.1?
- Linear in parameters
- model must be correctly specified
- error term is additive
Describe MLR.2
- Random Sampling
- observations in the sample are randomly selected from the population
Describe MLR.3
- No Perfect Collinearity
- must be variation on all of the independent variables
- there are no exact relationships among the independent variables
Describe MLR.4
-Zero conditional mean
- value of the explanatory variables must contain no info about the mean of the unobserved factors
What is meant by an endogenous explanatory variable?
- an explanatory variable that is correlated with the error term
- endogeneity is a violation of MLR.4
What is meant by an exogenous explanatory variable?
- explanatory variable is uncorrelated with error term
- if all explanatory variables are exogenous MLR.4 holds
What is simultaneity?
When causality doesn’t just run from the regressors to the regressand but also in the other direction
Interpret unbiasedness
It is an average property in repeated samples.
- on average the estimated coefficient will equal the true value
What is omitted variable bias?(OVB)
- occurs when one or more explanatory variables are correlated with a relevant omitted variable subsumed in the error term)
- results in biased estimated coefficients
- more generally, in MLR if omitted variable is correlated with only one explanatory variable all estimates coefficients are likely to be biased
What is MLR.5?
Homescedasticity
- the value of the explanatory variables must contain no info about the variance of the unobserved factors
What is the effect of a high error variance on the sampling variance?
- High error variance leads to an increase in sampling variance.
- As a result a large error variance makes estimates more imprecise.
- error variance does not decrease with sample size
What is the effect of a higher sample variation in the explanatory variable on the OLS estimates?
- More sample variation leads to more precise estimates
- total sample variation increases with sample size
Effect of a high Rj-Squared.
High Rj-Squared means there is a strong correlation between explanatory variable Xj and the other explanatory variables
- higher Rj-Squared increases variance of estimate Bj
- problem of almost linearly dependent explanatory variables is called multi Collinearity
- Occurs as Rj approaches 1
Is multi Collinearity a violation of our assumptions?
- does not violate any of our assumptions
- but troublesome
How can we lessen effect of multi Collinearity?
Increase sample size
What is Gauss-Markov theorem?
Under assumptions MLR.1-MLR.5 Bo, B1……. Are the Best Linear Unbiased Estimators (BLUE) of the true population values
How to tell if a regression suffers from multi Collinearity.
- VIF > 10 indicates m/c in regression
- t-tests reject the individual significance of several coefficients but overall R-Squared of regression is still high
- Estimates are imprecise
- Estimates will be sensitive to changes in specification of the model
