Formulas for Econometrics Part
Ovidijus Stauskas
Many of the formulas used and needed in this course you can find in the Statistics Formulas sheet. However, here are a few additional relevant formulas that we will use only for econometrics.
1. b1 = yn − b2 x n . Least squares estimator of B1 , where x n and yn are the sample averages of observations for x and y variables.
2. b2 =
∑in=1 ( xi − x n )(yi −yn )
∑in=1 ( xi − x n )2
=
1
n −1
vided by sample variance).
3. b2 =
∑in=1 ( xi − x n )(yi −yn )
∑iN=1 ( xi − x n )2
∑in=1 ( xi − x n )(yi −yn )
.
n
1
2
n −1 ∑ i =1 ( x i − x n )
n
1
n −1 ∑ i =1 ( x i − x n ) y i
N
1
2
n −1 ∑ i =1 ( x i − x n )
=
Least squares estimator of B2 (sample covariance di-
. Alternative (and perhaps surprising!) formula for the
estimator of B2 for the simple constant and one variable model.
n
1
2
4. b
σ2 = n−
2 ∑i =1 ei , where ei = yi − b1 − b2 xi . Estimator of regression equation variance (i.e. estimator
of variance of ui ). This is for a simple model with B1 and B2 .
∑in=1 ei2 , where ei = yi − b1 − b2 x2,i − . . . − bk xk,i . For the model with B1 , . . . , Bk .
r
1
∑n x2
\
= Var (b1 ) = b
σ ∑nn (xi=−1 xi )2 : variance estimator for B1 from the simple B1 , B2 model.
5. b
σ2 =
6. s2b1
1
n−k
i =1
\
7. s2b2 = Var
(b2 ) =
i
b
σ2
.
∑in=1 ( xi − x n )2
n
Variance estimator for b2 from the simple B1 , B2 model.
8. TSS (total sum of squares): ∑in=1 (yi − yn )2 .
9. ESS (explained sum of squares): ∑in=1 (yi − ŷi )2 , where ŷi = b1 + b2 x2,i + . . . + bk xk,i (obviously, k = 2
in a simple constant and one explanatory variable model).
10. RSS (residual sums of squares): ∑in=1 ei2 , where ei is defined as above, depending if we have a simple
2 variable or k variable model.
11. TSS = ESS + RSS (an important result showing that TSS can be split into two parts).
12. R2 =
ESS
TSS
=
TSS− RSS
TSS
= 1−
RSS
TSS ,
where the second equality follows straight from point 11.
2
1
2
13. R = 1 − (1 − R2 ) nn−
−k : adjusted R .
\
14. s2bj = Var
(b j ) =
b
σ2
1
,
∑in=1 ( xi − x n )2 1− R2j
where R2j is the R2 from the regression where x j,i is the dependent
variable and the rest of the variables are used as predictors.
15. F =
RSSr − RSSu
m
RSSu
n−k
. This is F statistic for joint hypotheses. Here, r stands for “restricted model”, u stands
for “unrestricted model”. Also, m is the number of restrictions; n and k stand for the sample size an
the number of coefficients in the unrestricted model.
1
16. F =
R2u − R2r
m
1− R2u
n−k
: an alternative formula for F statistic. The letters have the same meaning as in the original
formulations.
17. F =
R2u
m
1− R2u
n−k
: a special case of F statistic when the restricted model involves no explanatory variables
(predictors) at all.
18. E(Yt ) = 1−δ ρ : expected value of AR(1) process, where δ is an intercept and ρ is the autoregressive
coefficient.
19. Var (Yt ) =
σ2
:
1− ρ2
variance of AR(1) process, where σ2 is the variance of the error term.
2
20. Cov(Yt , Yt−s ) = ρs 1−σ ρ2 : covariance between two different time points in the AR(1) process, where
the “time step” (or, difference in time) is s.
2