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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 


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Definition
∑e_{t}=0 (sum of errors=0)
∑e_{t}x_{t}=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 


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Definition


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Definition


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Definition
Best linear unbiased estimator.
If A5 it is consistent 


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Definition


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Definition
limit_{n→∞}Bias(β^{^})=0
limit_{n→∞}Var(β^{^})=0 


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Definition
1(SSE/(nk)/(SST/(n1))
This is a measure of the overall goodness of fit that has the property that adding an additional variable will only increase the adjusted R^{2 } if the absolute value of the tstatistic 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 tstatistic (θ^{^}θ)/(s_{θ^}) is not an exact tstatistic, but it may have an asymptotic normal distribution. 


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Definition
In timeseries data, if the error terms between different times are correlated, this violates A4 and the OLS estimator would not be BLUE. 


Term

Definition
Source

Sum of squared errors

Degrees of freedom

Mean squared error

Model

SSR

k1

SSR/(k1)

Error

SSE

nk

SSE/(nk) =s^{2}

Total

SST

n1

SST/(n1)




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Definition
Variables that take the values of 0 or 1. 


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Definition
To test joint hypothesis.
[(SSE*SSE)/r]/[SSE/(nk)]~F(r, nk)
where r=(nk)*(nk) or the number of equal signs in constraint
* meaning the restrained model
Can also look like [(R^{2*}R^{2})/r]/[(1R^{2})/(nk)]~F(r, nk) 


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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(s_{θ^}), θ^{^}+t(s_{θ^}). Where s_{θ^ }is the standard error. 


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Definition
a data set corresponding to sampling a population at a given point in time 


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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) 


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Definition
Minimum variance estimator. 


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Definition
when a explanatory variable is correlated with the error term. Causes OLS to be biased and inconsistent. 


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Definition
Can test multiple hypothesis. Most common is to test that all variables have no explanatory power. The test is [SSR/(k1)/[SSE/(nk)] or [R^{2}/(k1)]/[(1R^{2})/(nk)]~F(k1, nk) 


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Definition
Expresses the price or ln(price) of something in terms of its attributes. i.e. pricehome=f(squarefootage, bedrooms, bathrooms, age, size,...) 


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Definition
Violation of A3 meaning that Var(ε_{t})=σ_{t}^{2}≠σ^{2}. OLS is not BLUE. 


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3 Characteristics of Instrumental Variables 

Definition
1) variable does not appear in equation
2) uncorrelated with error
3) correlated with endogenous regressor 


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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 


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


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Definition
E(Yμ)^{4}/σ^{4}
This is a measure of peakedness and tail thickness. Kurtosis value for normal is 3. 


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Definition
Test validity of joint hypothesis. LR=2(ll*)~χ^{2}(r) where r is the number of equal signs in hypothesis. 


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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

Definition
Model for binary dependent variables. Pr(Y=1X)=the integral from ∞ to Xβ of f(s)ds =F(Xβ). f(s) is the loglogistic pdf (e^{s}/(1+e^{s})^{2}).
Marginal effect is β_{i}F(X_{i}β). 


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Definition
E(Y)=μ meaning the measure of central tendency. 


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Definition
nonzero correlations between explanatory variables in model. Increased collinearity makes the estimators less precise with significant Ftest, and insignificant ttests. 


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Definition
Timeseries data. Repeated crosssectional data over time. Balanced if the same number of data appears each time. 


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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. 


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Definition
Binary dependent variable model. Pr(Y=1X)=integral from ∞ to Xβ of f(s) ds=F(Xβ) where f(s)=(e^{s^}^{2/2}/√2π). The marginal effect is βF(Xβ). 


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Definition
Probability of a higher value of the test statistic under the null hypothesis. Smaller pvalues mean you can reject the hypothesis. 


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Qualitative Response Model 

Definition


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Definition
SSR/SST or 1SSE/SST. It is the fraction of the model that is explained by the independent variables. 


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Definition
representation of model where each dependent variable is expressed in terms of explanatory variables 


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Definition
E(Yμ)^{3}/σ^{3}
Measures symmetry or asymetry. Positive skewness means long thick tail to right. 


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Definition
data mining. No theory behind it so it is a bad word in economics. 


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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. 


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Definition
tests hypothesis that coefficient is equal to a number. Test is (θ^{^}θ)/s_{θ}~t(nk). Also used to test that a linear combination of coefficients is equal to a number. 


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Definition
If Y~N[μ_{y}, ∑_{y}] then Z=AY~N[Aμ_{y}, A∑_{y}A']
We use this to determine the OLS distribution, predictions, and linear combinations of estimators. 


Term
Variance Inflation Factor 

Definition
1/(1ρ_{i.})
Measures collinearity. ρ_{i. }is defined as the R^{2} obtained from regression the i^{th }X on the other X's. 


Term
estimator and distribution of β_{1} 

Definition
Ybarβ_{2}^{^}Xbar
~N[β_{1}, σ^{2}/n +Xbar^{2}σ^{2}/∑(X_{t}Xbar)^{2} 


Term
Estimator and Distribution of β_{2}^{^} 

Definition
Cov(X, Y)/Var(X)
~N[β_{2}, σ^{2}/∑(X_{t}Xbar)^{2}]



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Estimator and Distribution of β (Matrix Form) 

Definition
(X'X)^{1}X'Y
~N[β, σ^{2}(X'X)^{1}] 


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Definition
σ^{2}/n +(X_{t}Xbar)^{2}σ^{2}/∑(X_{t}Xbar)^{2}
σ^{2}X_{0}(X'X)^{1}X'_{0}



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Definition


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Definition
(nk)s^{2}_{β}/σ^{2}_{β}~χ^{2}(nk) where s^{2}=SSE/(n2) 


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Definition
Includes lagged dependent variables. Can use the Koyck method for estimating. 


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Definition
Regress Y on lagged Y's and lagged X's. If the coefficients of the X's are nonzero then, X grangercauses Y. The test can be done using an F or Chow test. 


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Definition
A plot of the correlation between a variable at different lag lengths. 


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Definition
Tests if the ΔY=μ+u_{t} is homoskedastic and uncorrelated. Null hypothesis is that you should use differences. 


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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. 


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Definition
Time series model that includes lagged explanatory variables. Can use Koyck and PDL to get rid of multicollinearity issues. 


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Koyck Distributive Lag Model 

Definition
Time series regression with lagged explanatory variables. Additional restriction that βi=λ^{i}β_{0}. 


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Polynomial Distributed Lag Model 

Definition
Time series regression model that includes lagged explanatory variables. Additional restriction that β_{i}=a_{0}+a_{1}i+a_{2}i^{2}+... 


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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. 


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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. 

