Econometrics, Econ 20
Professor Ethan Lewis
Formula Sheet and Tables
This formula sheet is for your reference on the exam. This is the only course material you
should reference during the exam.
1. Mean and Variance Rules. Suppose 𝑋 & 𝑌 are random variables, and 𝑎 & 𝑏 & 𝑐 are
constants. Then:
Mean: 𝐸[𝑎𝑋+𝑏𝑌+𝑐]
= 𝑎𝐸[𝑋]+𝑏𝐸[𝑌]+𝑐
Variance: 𝑉𝑎𝑟[𝑎𝑋+𝑏𝑌+𝑐]
= 𝑎) 𝑉𝑎𝑟[𝑋] + 𝑏 ) 𝑉𝑎𝑟[𝑌] + 2𝑎𝑏𝐶𝑜𝑣(𝑋, 𝑌)
If X & Y are independent
= 𝑎) 𝑉𝑎𝑟[𝑋] + 𝑏 ) 𝑉𝑎𝑟[𝑌]
2. In a bivariate regression Yi = β0 + β1Xi + ui OLS estimates of coefficients are given by:
bˆ0 = Y - bˆ1 X
bˆ1 = Cov( X i , Yi ) Var( X i )
3. If the multivariate model Yi = b 0 + b1 X 1i + b 2 X 2i + ... + b K X Ki + ui is estimated by
OLS, then the “jth” slope (where 0< j £K) is given by
bˆ j = Cov (r ji , Yi ) Var (r ji )
Where rji are the residuals from an OLS regression of X ji on the rest of the X’s in
the regression (that is, bˆ is the slope from a bivariate regression of Y on r ). This
i
j
ji
is also known as the Frisch-Waugh theorem. (The intercept, as always, is given by
bˆ0 = Y - bˆ1 X 1 - bˆ2 X 2 - ... - bˆ K X K .)
4. If the multivariate model Yi = b 0 + b1 X 1i + b 2 X 2i + ... + b K X Ki + ui is estimated by
OLS, then the standard error on the “jth” slope (where 0< j £K) is given by:
( )
se bˆ j = s
(N - 1) var(x j )(1 - R 2j )
…where s is the root MSE, N is the sample size, and R 2j is the R2 from an OLS
regression of xj on the rest of the X’s in the regression.
5. The F-statistic for testing linear restrictions on coefficients is given by:
F - stat =
Formula Sheet
(R
(1 - R
)
2
- Rrestricted
Q
(N - K - 1)
2
unrestricted
2
unrestricted
)
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Quiz #5
Econometrics, Econ 20
Professor Ethan Lewis
…where “Q” is the number restrictions, and “K” is the total #of X’s (that is, in the
unrestricted regression).
6. If the true population model is given by Yi = b 0 + b1 X 1i + b 2 X 2i + controls + ui and
~
~
you estimate Yi = b 0 + b1 X 1i + controls + u~i and X2 and X1 are linearly related by the
~
relationship X 2i = d 0 + d 1 X 1i + controls + ei , then your OLS estimate of b1 will
satisfy:
~ˆ
b1 = bˆ1 + bˆ2dˆ1
…where coefficients with a “hat” are estimated by OLS.
7. In a test of a single null hypothesis that a given bj = 0, it is always the case that the
F-statistic for this test exactly equals the square of the corresponding t-statistic
(F-stat = t-stat2). This is also true of the relevant critical values.
8. OLS estimates of 𝑌5 = β8 + β9 𝑋5 + u5 where 𝑋5 suffers from classical measurement
error such that 𝑋5 = 𝑋5∗ + 𝑒5 , where 𝑋5∗ is the “true” X and 𝑒5 is the reporting error
(with 𝑒5 uncorrelated with all variables besides 𝑋5 ), we have that, ignoring other
B9 = 𝜆𝛽9 , where the attenuation factor 𝜆 =
sources of omitted variables bias, 𝑝𝑙𝑖𝑚 𝛽
∗ [
∗
∗
𝑣𝑎𝑟(𝑋5 )⁄ 𝑣𝑎𝑟(𝑋5 ) + 𝑣𝑎𝑟(𝑒5 )] = 𝑣𝑎𝑟(𝑋5 )⁄𝑣𝑎𝑟(𝑋5 ).
9. In a bivariate regression Yi = β0 + β1Xi + ui estimated using Zi as an instrument for Xi,
HIJ(K,L)
HIJ(K,L) HIJ(M,L)
the slope estimate is given by 𝛽E FG =
=
P
.
HIJ(M,L)
JNO(L)
JNO(L)
10. When the validity conditions for valid instrumental variables (IV) estimates are met,
the standard error on the difference between IV and OLS slope estimates is given by
)
)
Q𝑠𝑒S𝛽EFG T − 𝑠𝑒S𝛽EVWX T where 𝑠𝑒S𝛽EFG T and 𝑠𝑒S𝛽EVWX T are, respectively, the standard
errors on IV and OLS slope estimates.
Statistical tables appear on the attached pages.
Formula Sheet
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Quiz #5
Econometrics, Econ 20
Formula Sheet
Professor Ethan Lewis
Page 3 of 4
Quiz #5
Econometrics, Econ 20
Professor Ethan Lewis
Degrees of Freedom
Denominator Numerator
Large
1
Large
2
Large
3
Large
4
Large
5
Large
6
Large
7
Large
8
Large
9
Large
10
Large
11
Large
12
Large
13
Large
14
Large
15
Large
16
Large
17
Large
18
Large
19
Large
20
Large
30
Large
40
Large
50
Large
60
Large
70
Large
80
Large
90
Large
100
Large
110
Large
120
Large
130
Large
140
Large
150
Large
160
Large
170
Large
180
Large
190
Large
200
…
Large
1000
Formula Sheet
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Significance Level
1%
5%
10%
6.637
3.842
2.706
4.607
2.997
2.303
3.784
2.606
2.084
3.321
2.373
1.945
3.019
2.215
1.848
2.804
2.099
1.775
2.641
2.011
1.717
2.513
1.939
1.671
2.409
1.881
1.632
2.323
1.832
1.599
2.249
1.790
1.571
2.187
1.753
1.546
2.132
1.721
1.525
2.083
1.693
1.505
2.040
1.667
1.488
2.002
1.645
1.472
1.967
1.624
1.458
1.935
1.605
1.445
1.907
1.588
1.432
1.880
1.572
1.421
1.697
1.459
1.342
1.592
1.394
1.295
1.523
1.350
1.263
1.473
1.318
1.240
1.435
1.293
1.222
1.404
1.274
1.207
1.379
1.257
1.195
1.358
1.244
1.185
1.340
1.232
1.176
1.325
1.222
1.169
1.311
1.213
1.162
1.299
1.205
1.156
1.288
1.197
1.151
1.279
1.191
1.146
1.270
1.185
1.141
1.262
1.180
1.137
1.254
1.175
1.134
1.248
1.170
1.130
1.108
1.075
1.058
Quiz #5