Econometrics Cheat Sheet
By Marcelo Moreno - King Juan Carlos University
The Econometrics Cheat Sheet Project
Basic concepts
Definitions
Econometrics - is a social science discipline with the
objective of quantify the relationships between economic
agents, test economic theories and evaluate and implement
government and business policies.
Econometric model - is a simplified representation of the
reality to explain economic phenomena.
Ceteris paribus - if all the other relevant factors remain
constant.
Data types
Cross section - data taken at a given moment in time, an
static photo. Order doesn’t matter.
Time series - observation of variables across time. Order
does matter.
Panel data - consist of a time series for each observation
of a cross section.
Pooled cross sections - combines cross section from dif-
ferent time periods.
Phases of an econometric model
1. Specification.
2. Estimation.
3. Validation.
4. Utilization.
Regression analysis
Study and predict the mean value of a variable (dependent
variable, y) regarding the base of fixed values of other vari-
ables (independent variables, x’s). In econometrics it is
common to use Ordinary Least Squares (OLS) for regres-
sion analysis.
Correlation analysis
Correlation analysis don’t distinguish between dependent
and independent variables.
Simple correlation measures the grade of linear associa-
tion between two variables.
r=Cov(x,y)
σx·σy=Pn
i=1((xi−x)·(yi−y))
√Pn
i=1(xi−x)2·Pn
i=1(yi−y)2
Partial correlation measures the grade of linear associa-
tion between two variables controlling a third.
Assumptions and properties
Econometric model assumptions
Under this assumptions, the OLS estimator will present
good properties. Gauss-Markov assumptions:
1. Parameters linearity (and weak dependence in time
series). ymust be a linear function of the β’s.
2. Random sampling. The sample from the population
has been randomly taken. (Only when cross section)
3. No perfect collinearity.
There are no independent variables that are constant:
Var(xj)= 0,∀j= 1,. . . , k .
There isn’t an exact linear relation between indepen-
dent variables.
4. Conditional mean zero and correlation zero.
a. There aren’t systematic errors: E(u|x1, . . . , xk) =
E(u)=0→strong exogeneity (a implies b).
b. There are no relevant variables left out of the model:
Cov(xj, u)=0,∀j= 1, . . . , k →weak exogeneity.
5. Homoscedasticity. The variability of the residuals is
the same for all levels of x:
Var(u|x1,. . . , xk) = σ2
u
6. No auto-correlation. Residuals don’t contain infor-
mation about any other residuals:
Corr(ut, us|x1, . . . , xk)=0,∀t=s.
7. Normality. Residuals are independent and identically
distributed: u∼N(0, σ2
u)
8. Data size. The number of observations available must
be greater than (k+ 1) parameters to estimate. (It is
already satisfied under asymptotic situations)
Asymptotic properties of OLS
Under the econometric model assumptions and the Central
Limit Theorem (CLT):
Hold 1 to 4a: OLS is unbiased. E( ˆ
βj) = βj
Hold 1 to 4: OLS is consistent. plim( ˆ
βj) = βj(to 4b
left out 4a, weak exogeneity, biased but consistent)
Hold 1 to 5: asymptotic normality of OLS (then, 7 is
necessarily satisfied): u∼
aN(0, σ2
u)
Hold 1 to 6: unbiased estimate of σ2
u. E(ˆσ2
u) = σ2
u
Hold 1 to 6: OLS is BLUE (Best Linear Unbiased Esti-
mator) or efficient.
Hold 1 to 7: hypothesis testing and confidence intervals
can be done reliably.
Ordinary Least Squares
Objective - minimize the Sum of Squared Residuals (SSR):
min Pn
i=1 ˆu2
i, where ˆui=yi−ˆyi
Simple regression model
y
x
β0
β1
Equation:
yi=β0+β1xi+ui
Estimation:
ˆyi=ˆ
β0+ˆ
β1xi
where: ˆ
β0=y−ˆ
β1x
ˆ
β1=Cov(y,x)
Var(x)
Multiple regression model
x2
y
x1
β0
Equation:
yi=β0+β1x1i+· · · +βkxki +ui
Estimation:
ˆyi=ˆ
β0+ˆ
β1x1i+· ·· +ˆ
βkxki
where:
ˆ
β0=y−ˆ
β1x1− · ·· − ˆ
βkxk
ˆ
βj=Cov(y,resid xj)
Var(resid xj)
Matrix: ˆ
β= (XTX)−1(XTy)
Interpretation of coefficients
Model Dependent Independent β1interpretation
Level-level y x ∆y=β1∆x
Level-log ylog(x) ∆y≈(β1/100)(%∆x)
Log-level log(y)x%∆y≈(100β1)∆x
Log-log log(y) log(x) %∆y≈β1(%∆x)
Quadratic y x +x2∆y= (β1+ 2β2x)∆x
Error measurements
Sum of Sq. Residuals: SSR = Pn
i=1 ˆu2
i=Pn
i=1(yi−ˆyi)2
Explained Sum of Squares: SSE = Pn
i=1(ˆyi−y)2
Total Sum of Sq.: SST = SSE + SSR = Pn
i=1(yi−y)2
Standard Error of the Regression: ˆσu=qSSR
n−k−1
Standard Error of the ˆ
β’s: se( ˆ
β) = pˆσ2
u·(XTX)−1
Mean Squared Error: MSE = Pn
i=1(yi−ˆyi)2
n
Absolute Mean Error: AME = Pn
i=1|yi−ˆyi|
n
Mean Percentage Error: MPE = Pn
i=1|ˆui/yi|
n·100
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