Metrics 2019 Lecture6 - HUANG, Jia-Ping · Linear relationship between X and Y • A school...

Econometrics 1 Lecture 6: Linear Regression (1)Linear regression with one regressor

⻩黄嘉平中国经济特区研究中⼼心讲师办公室：⽂文科楼2613 E-mail: huangjp@szu.edu.cnTel: (0755) 2695 0548Website: https://huangjp.com

The linear regression model

Linear relationship between X and Y

• A school district cuts the size of its elementary school classes. What is the effect on its students’ test score?

• This question is about the unknown effect of changing one variable, X (class size), on another variable, Y (student test score).

• Linear regression (with one regressor) is a model investigating the linear relationship between X and Y.

Class size and test score

• Relative change, or the effect of changing X on Y:This is the definition of the slope of a straight line relating test scores and class size:

bClassSize =change in TestScorechange in ClassSize

=DTestScoreDClassSize

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DTestScore = bClassSize ⇥DClassSize<latexit sha1_base64="Pwd0dKBStC0yRX0HphMD+btkc9o=">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</latexit><latexit sha1_base64="Pwd0dKBStC0yRX0HphMD+btkc9o=">AAACVHicbVBNSwMxEE23ftT6VfXoJVgEDyK7IuhFKFTBY6X2A7qlZNOpBpPdJZkV69Jf5K/xIqh/xIMHs7WorQ4EHu/NzMu8IJbCoOu+5Jz83PzCYmGpuLyyurZe2thsmijRHBo8kpFuB8yAFCE0UKCEdqyBqUBCK7itZnrrDrQRUXiFwxi6il2HYiA4Q0v1Suf+GUhk1N+nV2CwziMN9JT6ASDrpZatSmZMXTzAiPooFBj6M/Gt9Upl98AdF/0LvAkok0nVeqV3vx/xREGIPFvS8dwYuynTKLiEUdFPDMSM37Jr6FgYMmvcTcfnjuiuZfp0EGn7QqRj9vdEypQxQxXYTsXwxsxqGfmf1klwcNJNRRgnCCH/MhokkmJEs+xoX2jgKIcWMK6F/SvlN0wzjjbhKZdATd2QZl4xqvtR0UblzQbzFzQPDzyLL4/KlcoktALZJjtkj3jkmFTIBamRBuHkkTyRV/KWe859OHln/qvVyU1mtshUOWufzHK0fg==</latexit><latexit sha1_base64="Pwd0dKBStC0yRX0HphMD+btkc9o=">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</latexit><latexit sha1_base64="Pwd0dKBStC0yRX0HphMD+btkc9o=">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</latexit>

TestScore = b0 +bClassSize ⇥ClassSize<latexit sha1_base64="5vQjGOBB/8/CSQ5bBtnEV2npaKU=">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</latexit><latexit sha1_base64="5vQjGOBB/8/CSQ5bBtnEV2npaKU=">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</latexit><latexit sha1_base64="5vQjGOBB/8/CSQ5bBtnEV2npaKU=">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</latexit><latexit sha1_base64="5vQjGOBB/8/CSQ5bBtnEV2npaKU=">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</latexit>

Incorporating other factors

• This relation may not hold for all districts. Therefore we must incorporate other factors influencing test scores.

• In a more general expression, ClassSize becomes X, and TestScore becomes Y.

TestScore = b0 +bClassSize ⇥ClassSize+other factors<latexit sha1_base64="nX4y9SdO78k3SkkgjWrSW1Iumg0=">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</latexit><latexit sha1_base64="nX4y9SdO78k3SkkgjWrSW1Iumg0=">AAACYXicbZDNTuswEIXd8N/LhQBLNhbVlZC4QglCgg1SJTYs+SsgNVXluBNqYceRPUGUKO/Fq7BhwwIegSVOiQQFZvXpzNhn5sSZFBaD4LHhTU3PzM7NLzT/LP5dWvZXVi+szg2HDtdSm6uYWZAihQ4KlHCVGWAqlnAZ3xxW/ctbMFbo9BxHGfQUu05FIjhDJ/X903OweMa1AXpAoxiQ9QO6VVMR/aeHkll7Ju6hpBEKBfZTqeYQ7rDQOARDE8ZRG1v2/VawHYyL/oSwhhap67jvv0UDzXMFKfLq724YZNgrmEHBJZTNKLeQMX7DrqHrMGVui14xvr2k/5wyoIl2/jpFOla/viiYsnakYjepGA7t914l/tbr5pjs9wqRZjlCyj+MklxS1LQKkg6EAY5y5IBxI9yulA+ZcSG4uCdcYjVxQ1F5ZajuyqaLKvwezE+42NkOHZ/sttrtOrR5sk42yCYJyR5pkyNyTDqEkwfyRF7Ia+PZW/B8b/Vj1GvUb9bIRHnr7zBBuY8=</latexit><latexit sha1_base64="nX4y9SdO78k3SkkgjWrSW1Iumg0=">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</latexit><latexit sha1_base64="nX4y9SdO78k3SkkgjWrSW1Iumg0=">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</latexit>

• The linear regression model with one regressor

Yi = �0 + �1Xi + ui

dependent variableindependent variable / regressor

error termcoefficients

• The linear regression model with one regressor

Yi = �0 + �1Xi + ui

population regression line / population regression function

A test score data in California: the STAR dataset

• The file caschool.xlsx

• The California Standardized Testing and Reporting (STAR) dataset (1998-1999).

• Average test scores on 420 districts in California.

• For details, see californiatestscores.docx

Average test score v.s. student-teacher ratio

• “testscr”: the average test score (of reading and math)

• “str”: the student-teacher ratio (No. of student / No. of teachers)

��

Estimation

Estimating the coefficients

• is an estimator of the population mean.

• Similarly, we need estimators of the coefficientsand .

• The ordinary least squares (OLS) estimators and are the ones that minimize

(Yi � b0 � b1Xi)2

�0 �1

How to determine the sample regression line ?�0 + �1X

��

sum of squared m

istakes

(Yi � b0 � b1Xi)2

The OLS estimator, predicted values, and residuals

• The OLS estimators of the slope and the intercept are

• The OLS predicted value:

• The residuals:

�1 =

Pni=1(Xi �X)(Yi � Y )Pn

i=1(Xi �X)2=

�0 = Y � �1X

Yi = �0 + �1Xi

ui = Yi � Yisample regression line/

sample regression function

��

� � �� \testscr = 698.9� 2.28⇥ str

Why use the OLS estimator

• OLS is the dominating method used in practice.

• Under certain assumptions, the OLS estimator is unbiased and consistent.

• With some further assumptions, the OLS estimator is also efficient among a class of unbiased estimators.

⇒ Gauss-Markov Theorem (Section 5.5)

For the definitions of unbiasedness, consistency, and efficiency, read Chapter 3.

Measures of fit

The R2

• The R2 — correlation of determination, the fraction of the sample variance of Yi explained by Xi.

• Recall that

Read Appendix 4.3 if you want to know why the second equality holds.

Pni=1(Yi � Y )2

= 1�Pn

i=1 u2iPn

i=1(Yi � Y )2= 1� SSR

(explained sum of squares)(total sum of squares)

(sum of squared residuals)

Yi = Yi + ui

A graphical explanation of SSR

OLS regressionSimple average of Yi

0 2 4 6 8 10

Y ~ X + N(0, 0.25)

R2 = 0.9637

0 2 4 6 8 10

Y ~ X + N(0, 1)

R2 = 0.9002

0 2 4 6 8 10

Y ~ X + N(0, 9)

R2 = 0.5426

0 2 4 6 8 10

Y ~ X + N(0, 100)

R2 = 0.03801

How to read R2

• R2 measures how well the OLS regression line fits the data.

• The value of R2 ranges between 0 and 1. A high R2 indicates that the regressor (Xi) is good at predicting Yi, while a low R2 indicates that the regressor (Xi) is not very good at predicting Yi .

• A low R2 does not imply that this regression is either “good” or “bad”, it does tell us that other important factors influence the dependent variable.

The standard error of the regression

• The standard error of the regression (SER) is an estimator of the standard deviation of the regression error .

• SER measures the magnitude of a typical deviation from the regression line.

• SER has the same units of the dependent variable.

ui<latexit sha1_base64="RRNwUYMp6e5EFMWEp9hMejARBmY=">AAACEXicbVDLSsNAFL2prxpfVZduBkvBVUmqoMuiG5cV7QPaUCbTSTt0JgkzE7GEfoIrQb/Fnbj1C/wUd07aLGrrgYHDOfdy7hw/5kxpx/m2CmvrG5tbxW17Z3dv/6B0eNRSUSIJbZKIR7LjY0U5C2lTM81pJ5YUC5/Ttj++yfz2I5WKReGDnsTUE3gYsoARrI10n/RZv1R2qs4MaJW4OSlDjka/9NMbRCQRNNSEY6W6rhNrL8VSM8Lp1O4lisaYjPGQdg0NsaDKS2enTlHFKAMURNK8UKOZuriRYqHURPhmUmA9UsteJv7ndRMdXHkpC+NE05DMg4KEIx2h7N9owCQlmk8MwUQycysiIywx0aadPym+mNqVRSELi7V4mtqmK3e5mVXSqlXd82rt7qJcv85bK8IJnMIZuHAJdbiFBjSBwBCe4RXerBfr3fqwPuejBSvfOYY/sL5+AXFsnhg=</latexit>

SER = su, where s2u =1

n� 2

n� 2<latexit sha1_base64="jp8AWKyouJT/lp8rSkiFjV2vOKQ=">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</latexit>

OLS regression in gretl

• From the menu:> Model > Ordinary least squares >

• Scripts:ols testscr const str

dependent variable regressors

Regression results in gretl

Model 1: OLS, using observations 1-420 Dependent variable: testscr

coefficient std. error t-ratio p-value --------------------------------------------------------- const 698.933 9.46749 73.82 6.57e-242 *** str −2.27981 0.479826 −4.751 2.78e-06 ***

Mean dependent var 654.1565 S.D. dependent var 19.05335 Sum squared resid 144315.5 S.E. of regression 18.58097 R-squared 0.051240 Adjusted R-squared 0.048970 F(1, 418) 22.57511 P-value(F) 2.78e-06 Log-likelihood −1822.250 Akaike criterion 3648.499 Schwarz criterion 3656.580 Hannan-Quinn 3651.693

The least square assumptions

The least squares assumptions

For the linear regression model it is assumed that:

1. The error term ui has conditional mean zero given Xi:

2. (Xi, Yi), i = 1, …, n, are i.i.d. draws from their joint distribution; and

3. Large outliers are unlikely: Xi and Yi have nonzero finite fourth moments.

Yi = �0 + �1Xi + ui, i = 1, . . . , n

E(ui | Xi) = 0 () corr(Xi,ui) = 0)<latexit sha1_base64="xNKTWs+hK47pB9itzYFyHqnw5u4=">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</latexit><latexit sha1_base64="xNKTWs+hK47pB9itzYFyHqnw5u4=">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</latexit><latexit sha1_base64="xNKTWs+hK47pB9itzYFyHqnw5u4=">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</latexit><latexit sha1_base64="xNKTWs+hK47pB9itzYFyHqnw5u4=">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</latexit>

Implication of E(ui | Xi) = 0

Linear regression is sensitive to outliers

Hypothesis tests and confidence intervals

Large-sample distributions of and

If the least square assumptions hold, then in large samples and have a jointly normal sampling distribution.

The large-sample distribution of is , where

�0<latexit sha1_base64="I93HTf8tBz33hLAMCi/LSJ2m88c=">AAAB9HicbVBNS8NAEN34WetX1aOXxSJ4KkkV9Fj04rGC/YAmlM122i7dbOLupFBCf4cXD4p49cd489+4bXPQ1gcDj/dmmJkXJlIYdN1vZ219Y3Nru7BT3N3bPzgsHR03TZxqDg0ey1i3Q2ZACgUNFCihnWhgUSihFY7uZn5rDNqIWD3iJIEgYgMl+oIztFLgDxlmfgjIpl23Wyq7FXcOukq8nJRJjnq39OX3Yp5GoJBLZkzHcxMMMqZRcAnTop8aSBgfsQF0LFUsAhNk86On9NwqPdqPtS2FdK7+nshYZMwkCm1nxHBolr2Z+J/XSbF/E2RCJSmC4otF/VRSjOksAdoTGjjKiSWMa2FvpXzINONocyraELzll1dJs1rxLivVh6ty7TaPo0BOyRm5IB65JjVyT+qkQTh5Is/klbw5Y+fFeXc+Fq1rTj5zQv7A+fwBxVWSGA==</latexit>

�1<latexit sha1_base64="LsbQjMyDSuqjNCwqvYo0d7NGx8I=">AAAB9HicbVBNS8NAEN34WetX1aOXxSJ4KkkV9Fj04rGC/YAmlM122i7dbOLupFBCf4cXD4p49cd489+4bXPQ1gcDj/dmmJkXJlIYdN1vZ219Y3Nru7BT3N3bPzgsHR03TZxqDg0ey1i3Q2ZACgUNFCihnWhgUSihFY7uZn5rDNqIWD3iJIEgYgMl+oIztFLgDxlmfgjIpl2vWyq7FXcOukq8nJRJjnq39OX3Yp5GoJBLZkzHcxMMMqZRcAnTop8aSBgfsQF0LFUsAhNk86On9NwqPdqPtS2FdK7+nshYZMwkCm1nxHBolr2Z+J/XSbF/E2RCJSmC4otF/VRSjOksAdoTGjjKiSWMa2FvpXzINONocyraELzll1dJs1rxLivVh6ty7TaPo0BOyRm5IB65JjVyT+qkQTh5Is/klbw5Y+fFeXc+Fq1rTj5zQv7A+fwBxtmSGQ==</latexit>

N(�1,�2�1)

<latexit sha1_base64="IgHs6sqob1e0M6MXcl8PopIjOtk=">AAACD3icbVBNS8NAEN34bf2KevSyWJQKUpIq6LHoxZMoWBWaGibbTbt088HuRCih/8CLf8WLB0W8evXmv3Fbg2j1wcDjvRlm5gWpFBod58OamJyanpmdmy8tLC4tr9ira5c6yRTjDZbIRF0HoLkUMW+gQMmvU8UhCiS/CnrHQ//qlistkvgC+ylvRdCJRSgYoJF8e/u0knsBRxj47i71tOhEcFPzc68L+G0Mdny77FSdEehf4hakTAqc+fa7105YFvEYmQStm66TYisHhYJJPih5meYpsB50eNPQGCKuW/nonwHdMkqbhokyFSMdqT8ncoi07keB6YwAu3rcG4r/ec0Mw8NWLuI0Qx6zr0VhJikmdBgObQvFGcq+IcCUMLdS1gUFDE2EJROCO/7yX3JZq7p71dr5frl+VMQxRzbIJqkQlxyQOjkhZ6RBGLkjD+SJPFv31qP1Yr1+tU5Yxcw6+QXr7RP+p5wD</latexit>

N(�0,�2�0)

<latexit sha1_base64="5Q7l6s/R7BaCn5q4iPGZzkuhyds=">AAACD3icbVBNS8NAEN34bf2KevSyWJQKUpIq6LHoxZMoWBWaGibbTbt088HuRCih/8CLf8WLB0W8evXmv3Fbg2j1wcDjvRlm5gWpFBod58OamJyanpmdmy8tLC4tr9ira5c6yRTjDZbIRF0HoLkUMW+gQMmvU8UhCiS/CnrHQ//qlistkvgC+ylvRdCJRSgYoJF8e/u0knsBRxj4zi71tOhEcFPzc68L+G0Mdny77FSdEehf4hakTAqc+fa7105YFvEYmQStm66TYisHhYJJPih5meYpsB50eNPQGCKuW/nonwHdMkqbhokyFSMdqT8ncoi07keB6YwAu3rcG4r/ec0Mw8NWLuI0Qx6zr0VhJikmdBgObQvFGcq+IcCUMLdS1gUFDE2EJROCO/7yX3JZq7p71dr5frl+VMQxRzbIJqkQlxyQOjkhZ6RBGLkjD+SJPFv31qP1Yr1+tU5Yxcw6+QXr7RP7gpwB</latexit>

�2�1

var[(Xi � µX)ui]

[var(Xi)]2<latexit sha1_base64="BrsIe+BEOZkrCz/wBR6aoUHMRbE=">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</latexit>

�2�0

var(Hiui)

[E(H2i )]

2, where Hi = 1�

E(X2i )

<latexit sha1_base64="aerE68CRU4+1G0AKJLA8lF/4uYs=">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</latexit>

Hypotheses concerning

• Two-sided hypotheses

• The t-statisticwhere

• The p-value

�1<latexit sha1_base64="pFWXS+Iii5Q23t0nR1cweqBHPQI=">AAAB7nicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT3mTI7IOZXiGEfIQXD4p49Xu8+TdOkj1oYkFDUdVNd1eQKmnIdb+dtfWNza3twk5xd2//4LB0dNw0SaYFNkSiEt0OuEElY2yQJIXtVCOPAoWtYHQ381tPqI1M4kcap+hHfBDLUApOVmp1AyTe83qlsltx52CrxMtJGXLUe6Wvbj8RWYQxCcWN6XhuSv6Ea5JC4bTYzQymXIz4ADuWxjxC40/m507ZuVX6LEy0rZjYXP09MeGRMeMosJ0Rp6FZ9mbif14no/DGn8g4zQhjsVgUZopRwma/s77UKEiNLeFCS3srE0OuuSCbUNGG4C2/vEqa1Yp3Wak+XJVrt3kcBTiFM7gAD66hBvdQhwYIGMEzvMKbkzovzrvzsWhdc/KZE/gD5/MH7ZyPTA==</latexit>

H0 : �1 = �1,0 vs. H1 : �1 6= �1,0<latexit sha1_base64="oAVJ7fcU+nWsM5yFAxFHbt/pZgE=">AAACMnicbVDLSgMxFM34tr6qLt0Ei+BCyowKiiCIbupOwdZCpwyZ9NaGZjJjcqdYhn6TG79EcKELRdz6EaZ1wOeBwMk555LcEyZSGHTdR2dsfGJyanpmtjA3v7C4VFxeqZk41RyqPJaxrofMgBQKqihQQj3RwKJQwmXYPRn6lz3QRsTqAvsJNCN2pURbcIZWCoqnlcA9oH4IyAKPHuYs87bcAfWpj3CDGe2ZMh1eK4H3lfUVXH+PB8WSW3ZHoH+Jl5MSyXEWFO/9VszTCBRyyYxpeG6CzYxpFFzCoOCnBhLGu+wKGpYqFoFpZqOVB3TDKi3ajrU9CulI/T6RsciYfhTaZMSwY357Q/E/r5Fie7+ZCZWkCIp/PtROJcWYDvujLaGBo+xbwrgW9q+Ud5hmHG3LBVuC93vlv6S2XfZ2ytvnu6Wj47yOGbJG1skm8cgeOSIVckaqhJNb8kCeyYtz5zw5r87bZ3TMyWdWyQ847x9/f6a5</latexit>

t =�1 � �1,0

SE(�1)<latexit sha1_base64="CNaffWS0zZ9NMPUgfkzNbn+oJuo=">AAACI3icbZDLSgMxFIYz9VbrrerSTbAIFbTMVEERhKIILivaC3TKkEkzbWjmQnJGKMO8ixtfxY0Lpbhx4buYXha29UDg4//P4eT8biS4AtP8NjJLyyura9n13Mbm1vZOfnevrsJYUlajoQhl0yWKCR6wGnAQrBlJRnxXsIbbvx35jWcmFQ+DJxhErO2TbsA9TgloyclfAb7GticJTewegcR2GZDUsfApHqOTWCdmmiaPd8UZ/zjFTr5glsxx4UWwplBA06o6+aHdCWnsswCoIEq1LDOCdkIkcCpYmrNjxSJC+6TLWhoD4jPVTsY3pvhIKx3shVK/APBY/TuREF+pge/qTp9AT817I/E/rxWDd9lOeBDFwAI6WeTFAkOIR4HhDpeMghhoIFRy/VdMe0QHBjrWnA7Bmj95EerlknVWKj+cFyo30ziy6AAdoiKy0AWqoHtURTVE0Qt6Qx/o03g13o2h8TVpzRjTmX00U8bPLxZvo1A=</latexit>

SE(�1) =q

�2�1, �2

1n�2

Pni=1(Xi �X)2u2

[ 1nPn

i=1(Xi �X)2]2<latexit sha1_base64="LEwcu97C78bjGpY3YxjXaeUNMVs=">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</latexit>

p -value = 2�(�|tact|)<latexit sha1_base64="aQNAl5Hgidmtn+tb+nu/ysiAOho=">AAACDXicbVDLSgNBEJz1bXxFPXoZjIKCht0o6EUIevEYwcRAdg2zk44ZMvtgpjcY1vyAF3/FiwdFvHr35t84Sfbgq6ChqOqmu8uPpdBo25/WxOTU9Mzs3HxuYXFpeSW/ulbTUaI4VHkkI1X3mQYpQqiiQAn1WAELfAlXfvds6F/1QGkRhZfYj8EL2E0o2oIzNFIzvxW7ey7CLab7PSYTGNATWnIrHbGzf4fXKeM4uNtt5gt20R6B/iVORgokQ6WZ/3BbEU8CCJFLpnXDsWP0UqZQcAmDnJtoiBnvshtoGBqyALSXjr4Z0G2jtGg7UqZCpCP1+0TKAq37gW86A4Yd/dsbiv95jQTbx14qwjhBCPl4UTuRFCM6jIa2hAKOsm8I40qYWynvMGUyMAHmTAjO75f/klqp6BwUSxeHhfJpFscc2SCbZIc45IiUyTmpkCrh5J48kmfyYj1YT9ar9TZunbCymXXyA9b7F1OgmxY=</latexit>

Confidence interval for

• The 95% confidence interval for is

�1<latexit sha1_base64="pFWXS+Iii5Q23t0nR1cweqBHPQI=">AAAB7nicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT3mTI7IOZXiGEfIQXD4p49Xu8+TdOkj1oYkFDUdVNd1eQKmnIdb+dtfWNza3twk5xd2//4LB0dNw0SaYFNkSiEt0OuEElY2yQJIXtVCOPAoWtYHQ381tPqI1M4kcap+hHfBDLUApOVmp1AyTe83qlsltx52CrxMtJGXLUe6Wvbj8RWYQxCcWN6XhuSv6Ea5JC4bTYzQymXIz4ADuWxjxC40/m507ZuVX6LEy0rZjYXP09MeGRMeMosJ0Rp6FZ9mbif14no/DGn8g4zQhjsVgUZopRwma/s77UKEiNLeFCS3srE0OuuSCbUNGG4C2/vEqa1Yp3Wak+XJVrt3kcBTiFM7gAD66hBvdQhwYIGMEzvMKbkzovzrvzsWhdc/KZE/gD5/MH7ZyPTA==</latexit>

[�1 � 1.96SE(�1), �1 + 1.96SE(�1)]<latexit sha1_base64="PXku9UK/o6pVl1DaFYFdNVFc27o=">AAACQXicdVDJSgNBFOxxjXEb9eilMQgRY5iJ4nILiuAxolkgM4SeTidp0rPQ/UYIQ37Ni3/gzbsXD4p49WJnOZhECx4UVfXofuVFgiuwrBdjbn5hcWk5tZJeXVvf2DS3tisqjCVlZRqKUNY8opjgASsDB8FqkWTE9wSret2rgV99YFLxMLiHXsRcn7QD3uKUgJYaZq3udAgkjseA9Bs2PsJ2/uIUOzl8d52dsA5y2MGT4cP/w27DzFh5awg8S+wxyaAxSg3z2WmGNPZZAFQQpeq2FYGbEAmcCtZPO7FiEaFd0mZ1TQPiM+Umwwb6eF8rTdwKpZ4A8FD9vZEQX6me7+mkT6Cjpr2B+JdXj6F17iY8iGJgAR091IoFhhAP6sRNLhkF0dOEUMn1XzHtEEko6NLTugR7+uRZUink7eN84fYkU7wc15FCu2gPZZGNzlAR3aASKiOKHtErekcfxpPxZnwaX6PonDHe2UETML5/AB+iq9k=</latexit>

Regression results in gretl

Heteroskedasticity and homoskedasticity

An example of heteroskedasticity

Definition

The error term is homoskedastic is the variance of the conditional distribution of given , is constant for i = 1, …, n and in particular does not depend on x.

Otherwise, the error term is heteroskedastic.

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Xi<latexit sha1_base64="3iZXTg4Ib3Bh78XuDxTfX/j8otU=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz00OnzfrniVt05yCrxclKBHI1++as3iFkaoTRMUK27npsYP6PKcCZwWuqlGhPKxnSIXUsljVD72fzUKTmzyoCEsbIlDZmrvycyGmk9iQLbGVEz0sveTPzP66YmvPYzLpPUoGSLRWEqiInJ7G8y4AqZERNLKFPc3krYiCrKjE2nZEPwll9eJa1a1buo1u4vK/WbPI4inMApnIMHV1CHO2hAExgM4Rle4c0Rzovz7nwsWgtOPnMMf+B8/gAxHo28</latexit>

var(ui | Xi = x)<latexit sha1_base64="Ctd4cmXJyRnVGxPQlfSvQva+cB0=">AAACB3icbZDLSsNAFIYnXmu9RV0KMliEuilJFXQjFN24rGAv0IQwmUzaoTNJmJkUS+jOja/ixoUibn0Fd76NkzYLbf1h4OM/5zDn/H7CqFSW9W0sLa+srq2XNsqbW9s7u+beflvGqcCkhWMWi66PJGE0Ii1FFSPdRBDEfUY6/vAmr3dGREgaR/dqnBCXo35EQ4qR0pZnHjkcqYHg2QiJSTX1KHQ4DWBXwxV8OPXMilWzpoKLYBdQAYWanvnlBDFOOYkUZkjKnm0lys2QUBQzMik7qSQJwkPUJz2NEeJEutn0jgk80U4Aw1joFyk4dX9PZIhLOea+7sy3lvO13Pyv1ktVeOlmNEpSRSI8+yhMGVQxzEOBARUEKzbWgLCgeleIB0ggrHR0ZR2CPX/yIrTrNfusVr87rzSuizhK4BAcgyqwwQVogFvQBC2AwSN4Bq/gzXgyXox342PWumQUMwfgj4zPH5+YmHo=</latexit>

Implications of homoskedasticity + least square assumptions

• The OLS estimators of coefficients are efficient among all estimators that are linear in . [BLUE]

• The standard errors of and reduce to simpler form, e.g.,where

Y1, . . . , Yn<latexit sha1_base64="b27HcVYdnYFeW0/UCccXTuCPn4k=">AAAB+HicbVDLTgJBEOzFF+ID1KOXicTEAyG7aKJHohePmAhIYLOZHWZhwuwjM70mSPgSLx40xquf4s2/cYA9KFjJJJWqrnRP+YkUGm3728qtrW9sbuW3Czu7e/vF0sFhS8epYrzJYhmrB59qLkXEmyhQ8odEcRr6krf90c3Mbz9ypUUc3eM44W5IB5EIBKNoJK9U7HhOhfT6MeoK6XhGKdtVew6ySpyMlCFDwyt9mTBLQx4hk1TrrmMn6E6oQsEknxZ6qeYJZSM64F1DIxpy7U7mh0/JqVH6JIiVeRGSufo7MaGh1uPQN5MhxaFe9mbif143xeDKnYgoSZFHbLEoSCXBmMxaIH2hOEM5NoQyJcythA2pogxNVwVTgrP85VXSqlWd82rt7qJcv87qyMMxnMAZOHAJdbiFBjSBQQrP8Apv1pP1Yr1bH4vRnJVljuAPrM8f/SOSAg==</latexit>

SE(�1) =q

�2�1

<latexit sha1_base64="QBCSkncQDCh3YKBfUzzeVgGIz4g=">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</latexit>

�2�1

=s2uPn

i=1(Xi �X)2<latexit sha1_base64="m+ZslpnZR37Cjep2H7wef34hjqo=">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</latexit>

Standard errors of

• Homoskedasticity-only standard error

• Heteroskedasticity-robust standard error (HC1)

SE(�1) =q

�2�1, �2

1n�2

Pni=1(Xi �X)2u2

[ 1nPn

i=1(Xi �X)2]2<latexit sha1_base64="LEwcu97C78bjGpY3YxjXaeUNMVs=">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</latexit>

SE(�1) =q

�2�1, �2

s2uPni=1(Xi �X)2

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In practice

• If the errors are heteroskedastic but the homoskedastic-only formulas are used⇒

• If the errors are homoskedastic but the heteroskedastic-robust formulas are used⇒

• Always use heteroskedastic-robust standard errors

t-statistic does not have a standard normal distribution, even in large samples

hypothesis tests and confidence intervals will be valid

Practice in gretl

Heteroskedasticity-robust estimation in gretl

• Settings for the whole scriptset force_hc on set hc_version 1 # 0 (the original White’s) is the defaultset robust_z on

• For single resressionols yvar xvar - -robust (you still need to set the HC version)

Settings in the preferences of gretl

Regression results in gretl (homoskedasticity-only)

Regression results in gretl(heteroskedasticity-robust with normal distribution)

Model 1: OLS, using observations 1-420 Dependent variable: testscr Heteroskedasticity-robust standard errors, variant HC1

coefficient std. error z p-value ------------------------------------------------------- const 698.933 10.3644 67.44 0.0000 *** str −2.27981 0.519489 −4.389 1.14e-05 ***

Mean dependent var 654.1565 S.D. dependent var 19.05335 Sum squared resid 144315.5 S.E. of regression 18.58097 R-squared 0.051240 Adjusted R-squared 0.048970 F(1, 418) 19.25943 P-value(F) 0.000014 Log-likelihood −1822.250 Akaike criterion 3648.499 Schwarz criterion 3656.580 Hannan-Quinn 3651.693

Exercises

1. Reproduce Table 4.1 using matrix.

2. Learn command gnuplot (or plot) and reproduce Figure 4.2 with appropriate titles and ranges of axes.

��

gnuplot testscr str --output=display --fit=none

Exercises (cont.)

3. Find the relation and differences among , , and by solving Exercises 4.9 and 4.12.

4. Learn Section 5.3.

�1<latexit sha1_base64="Xz1zwHx1orx4p94ggd9OvgKLRQo=">AAACG3icbVDLSsNAFJ3UV42vqks3wVJwVZIq6LLgxmUF+4AmlMl00g6dSeLMjVhCvsOVoN/iTty68FPcOWmz6MMDFw7n3Mu5HD/mTIFt/xiljc2t7Z3yrrm3f3B4VDk+6agokYS2ScQj2fOxopyFtA0MOO3FkmLhc9r1J7e5332iUrEofIBpTD2BRyELGMGgJc8dY0hdnwLOBs6gUrXr9gzWOnEKUkUFWoPKrzuMSCJoCIRjpfqOHYOXYgmMcJqZbqJojMkEj2hf0xALqrx09nRm1bQytIJI6gnBmqmLFykWSk2FrzcFhrFa9XLxP6+fQHDjpSyME6AhmQcFCbcgsvIGrCGTlACfaoKJZPpXi4yxxAR0T0spvsjM2qKQh8UgnjNTd+WsNrNOOo26c1lv3F9Vm82itTI6Q+foAjnoGjXRHWqhNiLoEb2gN/RuvBofxqfxNV8tGcXNKVqC8f0HcniiVg==</latexit>

rXY<latexit sha1_base64="Wb38CE1X67JdJu3WSLX7KzpNHG4=">AAACFHicbVDLSsNAFJ34rPVVdelmsBRclaQKuiy4cVnBPqQNZTKdtGNnkjBzI5aQf3Al6Le4E7fu/RR3Ttos+vDAwOGcezl3jhcJrsG2f6y19Y3Nre3CTnF3b//gsHR03NJhrChr0lCEquMRzQQPWBM4CNaJFCPSE6ztjW8yv/3ElOZhcA+TiLmSDAPuc0rASC3VTzoPab9Utqv2FHiVODkpoxyNfum3NwhpLFkAVBCtu44dgZsQBZwKlhZ7sWYRoWMyZF1DAyKZdpPptSmuGGWA/VCZFwCeqvMbCZFaT6RnJiWBkV72MvE/rxuDf+0mPIhiYAGdBfmxwBDi7Ot4wBWjICaGEKq4uRXTEVGEgiloIcWTabEyL2RhEcjntGi6cpabWSWtWtW5qNbuLsv1et5aAZ2iM3SOHHSF6ugWNVATUfSIXtAberderQ/r0/qaja5Z+c4JWoD1/Qf1yZ9x</latexit>

R2<latexit sha1_base64="F4Yyi8spA0dpmkFl00qVOLIjZy0=">AAACEXicbVDLSgMxFM3UVx1fVZdugqXgqsxUQZcFNy7row9ox5JJM21okhmSjFiG+QRXgn6LO3HrF/gp7sy0s+jDA4HDOfdybo4fMaq04/xYhbX1jc2t4ra9s7u3f1A6PGqpMJaYNHHIQtnxkSKMCtLUVDPSiSRB3Gek7Y+vM7/9RKSioXjQk4h4HA0FDShG2kj3d4+1fqnsVJ0p4Cpxc1IGORr90m9vEOKYE6ExQ0p1XSfSXoKkppiR1O7FikQIj9GQdA0ViBPlJdNTU1gxygAGoTRPaDhV5zcSxJWacN9McqRHatnLxP+8bqyDKy+hIoo1EXgWFMQM6hBm/4YDKgnWbGIIwpKaWyEeIYmwNu0spPg8tSvzQhYWaf6c2qYrd7mZVdKqVd3zau32olyv560VwQk4BWfABZegDm5AAzQBBkPwAt7Au/VqfVif1tdstGDlO8dgAdb3H9iznbs=</latexit>

References

1. Stock, J. H. and Watson, M. M., Introduction to Econometrics, 3rd Edition, Pearson, 2012.

2. Gretl User’s Guide

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