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Econometrics Midterm Definitions
Beginning Econometrics Definitions
60
Economics
Undergraduate 3
10/15/2011

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Cards

Term
5 Properties of OLS estimators
Definition

1) β's are unbiased

2) β's are consistent

3) β's are minimum variance of any unbiased estimator

4) MLE=OLS

5) β's are normally distributed

Term
OLS normal equations
Definition

∑et=0 (sum of errors=0)

∑etxt=0 (sum of (errors*x's)=0

In matrix form, X'e=0 or X'Xβ^=X'Y.

This means the predicted Y's and the errors are orthogonal.

 

Term
Properties of OLS estimators if A1 fails
Definition

1) β's are unbiased

2) β's are consistent

3) β's are NOT minimum variance of any unbiased estimator, but ARE minimum variance of linear unbiased estimators (BLUE)

4) OLS≠MLE

5) NOT normally distributed, but approximately are

Term
When does OLS=MLE?
Definition
When A1-A5 hold.
Term
When does OLS=BLUE?
Definition
If A2-A5 hold.
Term
Properties of BLUE
Definition

Best linear unbiased estimator.

If A5 it is consistent

Term
Biased
Definition
When E(β^)=E(β)
Term
Consistent
Definition

limitn→∞Bias(β^)=0

limitn→∞Var(β^)=0

Term
Adjusted R2
Definition

1-(SSE/(n-k)/(SST/(n-1))

This is a measure of the overall goodness of fit that has the property that adding an additional variable will only increase the adjusted R2  if the absolute value  of the t-statistic is greater than 1

Term
Asymptotic Distribution (asymptotic chi square)
Definition
f(θ^) is the asymptotic distribution of θ^ if  the exact distribution of θ^ approaches f(θ^) as n increases. The actual distribution can equal the asymptotic distribution. For example, if the errors aren't distributed normally, the t-statistic (θ^-θ)/(sθ^) is not an exact t-statistic, but it may have an asymptotic normal distribution.
Term
Autocorrelation
Definition
In time-series data, if the error terms between different times are correlated, this violates A4 and the OLS estimator would not be BLUE.
Term
ANOV table
Definition

Source

Sum of squared errors

Degrees of freedom

Mean squared error

Model

SSR

k-1

SSR/(k-1)

Error

SSE

n-k

SSE/(n-k) =s2

Total

SST

n-1

SST/(n-1)

Term
Binary/Dummy Variable
Definition
Variables that take the values of 0 or 1.
Term
Chow Test
Definition

To test joint hypothesis.

[(SSE*-SSE)/r]/[SSE/(n-k)]~F(r, n-k)

where r=(n-k)*-(n-k) or the number of equal signs in constraint

* meaning the restrained model

Can also look like [(R2*-R2)/r]/[(1-R2)/(n-k)]~F(r, n-k)

Term
Confidence Interval
Definition
a rule used to construct a random interval so that a certain percentage of all data sets, determined by the confidence level, yields an interval that contains the population value. CI for θ=(θ^-t(sθ^), θ^+t(sθ^). Where sθ^ is the standard error.
Term
Cross Sectional Data
Definition
a data set corresponding to sampling a population at a given point in time
Term
Dummy Variable Trap
Definition
With binary variables, you can only include an intercept if you don't include one of the binary options. (If variable can be 0, 1, or 2 you can eithe include two of those choices with an intercept or all three with no intercept)
Term
Efficiency
Definition
Minimum variance estimator.
Term
endogenous regressor
Definition
when a explanatory variable is correlated with the error term. Causes OLS to be biased and inconsistent. 
Term
 Most common F-test
Definition
Can test multiple hypothesis. Most common is to test that all variables have no explanatory power. The test is [SSR/(k-1)/[SSE/(n-k)] or [R2/(k-1)]/[(1-R2)/(n-k)]~F(k-1, n-k)
Term
Hedonic pricing model
Definition
Expresses the price or ln(price) of something in terms of its attributes. i.e. pricehome=f(squarefootage, bedrooms, bathrooms, age, size,...)
Term
heteroskedasticity
Definition
Violation of A3 meaning that Var(εt)=σt2≠σ2. OLS is not BLUE.
Term
3 Characteristics of Instrumental Variables
Definition

1) variable does not appear in equation

2) uncorrelated with error

3) correlated with endogenous regressor

Term
Instrumental Variables Explanation
Definition

To solve endogenous regressor problem. Use instrumental variable Z which satisfies 3 characteristics and then it will be consistent again. 

IV=OLS if Z=X

Term
interaction terms
Definition
A regressor where two explanatory variables are multiplied together. The marginal impact of one independent variable depends on another explanatory variable.
Term
kurtosis
Definition

E(Y-μ)44

This is a measure of peakedness and tail thickness. Kurtosis value for normal is 3.

Term
Likelihood Ratio Test
Definition
Test validity of joint hypothesis. LR=2(l-l*)~χ2(r) where r is the number of equal signs in hypothesis.
Term
Linear Probability Model
Definition
When Y is a binary dependent variable and is estimated with OLS. There are 3 problems with this. 1) violates A1, 2) violates A3, 3) it is possible for predictions to be outside of (0, 1)
Term
Logit model
Definition

Model for binary dependent variables. Pr(Y=1|X)=the integral from -∞ to Xβ of f(s)ds =F(Xβ). f(s) is the log-logistic pdf (es/(1+es)2).

Marginal effect is βiF(Xiβ).

Term
Mean
Definition
E(Y)=μ meaning the measure of central tendency.
Term
multicollinearity
Definition
non-zero correlations between explanatory variables in model. Increased collinearity makes the estimators less precise with significant F-test, and insignificant t-tests. 
Term
Panel Data
Definition
Time-series data. Repeated cross-sectional data over time. Balanced if the same number of data appears each time. 
Term
PDF
Definition
Probability Density Function. Satisfies the conditions that f(s)≥0 and that the integral of f(s) from -∞ to ∞ is 1. Also the Pr(a<Y<b)=integral from a to b of f(s)=F(b)-F(a) which is the CDF.
Term
Probit Model
Definition
Binary dependent variable model. Pr(Y=1|X)=integral from -∞ to Xβ of f(s) ds=F(Xβ) where f(s)=(e-s^2/2/√2π). The marginal effect is βF(Xβ).
Term
P-value
Definition
Probability of a higher value of the test statistic under the null hypothesis. Smaller p-values mean you can reject the hypothesis. 
Term
Qualitative Response Model
Definition
Probit or Logit models.
Term
R2
Definition
SSR/SST or 1-SSE/SST. It is the fraction of the model that is explained by the independent variables. 
Term
Reduced Form
Definition
representation of model where each dependent variable is expressed in terms of explanatory variables
Term
Skewness
Definition

E(Y-μ)33

Measures symmetry or asymetry. Positive skewness means long thick tail to right. 

Term
Stepwise Regression
Definition
data mining. No theory behind it so it is a bad word in economics. 
Term
Structural Representation of a Model
Definition
This is a mathematical representation of the relationship between economic variables implied by economic theory and may include endogenous regressors (endogenous variables) along with exogenous variables on the right hand side of the equations.
Term
T-statistic
Definition
tests hypothesis that coefficient is equal to a number. Test is (θ^-θ)/sθ~t(n-k). Also used to test that a linear combination of coefficients is equal to a number.
Term
"Useful" Theorem
Definition

If Y~N[μy, ∑y] then Z=AY~N[Aμy, A∑yA']

We use this to determine the OLS distribution, predictions, and linear combinations of estimators.

Term
Variance Inflation Factor
Definition

1/(1-ρi.)

Measures collinearity. ρi. is defined as the R2 obtained from regression the ith X on the other X's.

Term
estimator and distribution of β1
Definition

Ybar-β2^Xbar

~N[β1, σ2/n +Xbar2σ2/∑(Xt-Xbar)2

Term
Estimator and Distribution of β2^
Definition

 

Cov(X, Y)/Var(X)

~N[β2, σ2/∑(Xt-Xbar)2]

 

Term
Estimator and Distribution of β (Matrix Form)
Definition

(X'X)-1X'Y

~N[β, σ2(X'X)-1]

Term
σ2yhat
Definition

σ2/n +(Xt-Xbar)2σ2/∑(Xt-Xbar)2 

σ2X0(X'X)-1X'0  

 

 

Term
σ2FE
Definition

 

σ22yhat

 

Term
Testing for σ2
Definition
(n-k)s2β2β2(n-k) where s2=SSE/(n-2)
Term
Autoregressive Model
Definition
Includes lagged dependent variables. Can use the Koyck method for estimating. 
Term
Granger Causality Test
Definition
Regress Y on lagged Y's and lagged X's. If the coefficients of the X's are nonzero then, X granger-causes Y. The test can be done using an F or Chow test.
Term
Correlogram
Definition
A plot of the correlation between a variable at different lag lengths.
Term
Dickey-Fuller Test
Definition
Tests if the ΔY=μ+ut is homoskedastic and uncorrelated. Null hypothesis is that you should use differences.
Term
Difference in Differences Model
Definition
Use when examining policies with two different time periods (before and after). The interaction term between the two tells you the D in D.
Term
Distributed Lag Models
Definition
Time series model that includes lagged explanatory variables. Can use Koyck and PDL to get rid of multicollinearity issues.
Term
Koyck Distributive Lag Model
Definition
Time series regression with lagged explanatory variables. Additional restriction that βiiβ0.
Term
Polynomial Distributed Lag Model
Definition
Time series regression model that includes lagged explanatory variables. Additional restriction that βi=a0+a1i+a2i2+...
Term
Poisson Regression
Definition
A regression model with an integer valued dependent variable which is distributed as a poisson variable with a mean value that is a linear function of the explanatory variables.
Term
Spurious Regression
Definition
When the independent variables show correlation but there is zero explanatory power. The regression is just showing that there is a trend. This can be fixed by adding a time variable.
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