3. Spatial Autocorrelation

Size: px
Start display at page:

Download "3. Spatial Autocorrelation"

Transcription

1 . Spatal Autocorrelato. Spatal autocorrelato ad spatal lag Noto of Spatal autocorrelato Basc property of spatally located data: The observatos,,, of a geo-refereced varable X are lkely related over space Tobler s frst la of geography: Everythg s related to everythg, but ear thgs are more related tha dstat thgs Stepha (Hepple, 978: Data of geographc uts are ted together lke buches of grapes, ot separate lke balls a ur. Defto: Spatal autocorrelato s a assessmet of the correlato of a varable referece to spatal locato of the varable.e. t s the correlato of a varable th tself over space. If the observatos,,, of a geo-refereced varable X dsplay terdepedece over space, the data are sad to be spatally autocorrelated.

2 Spatal autocorrelato (Clff ad Ord,97: If the presece of some quatty a couty (samplg ut makes ts presece eghbourg coutes (samplg uts more or less lkely, e say that the pheomeo ehbts spatal autocorrelato. Spatal autocorrelato (Clff ad Ord, 98: If there s systematc spatal varato the varable, the the pheomeo beg studed s sad to ehbt spatal autocorrelato. Characterstcs of spatal autocorrelato - If there s ay systematc patter the spatal dstrbuto of a varable X, t s sad to be spatally autocorrelated - If earby or eghbourg areas are more alke, ths s postve spatal autocorrelato - Negatve autocorrelato descrbes patters hch eghborg areas are ulke (e.g. by competto - Radom patters ehbt o spatal autocorrelato

3 Spatal patter of a geo-refereced varable Radom patter - Values observed at a locato do ot deped o values at a eghbourg locato - Observed spatal patter of values s equally lkely as ay other spatal patter Spatal clusterg - Smlar values ted to cluster space - Neghbourg values are much alke

4 Spatal lag Spatal autocorrelato measures make use of the cocept of the spatal lag that has some aalogy ad dffereces to the cocept of the tme lag Lag operator L tme-seres aalyss: Shfts observatos of a varable Y oe or more perods back tme Frst-order lag: Ly t = y t- Secod-order lag: L y t = y t- k-th order lag: L k y t = y t-k, k=,, Spatal lag operator L: Relates a varable X at oe ut space to the observatos of that varable other spatal uts Dffereces betee lags tme-seres aalyss ad spatal ecoometrcs: Because tme s udrectoal, the applcato of lag operator L tme-seres aalyss s straghtforard. I spatal arragemets, a umber of shfts dfferet drectos are possble. Sce space s characterzed by multdrectoalty, frst-order lags, secod-order lags, lack straghtforardess. 4

5 Soluto of the problem of multdrectoalty: Use of the eghted sum of all values belogg to a gve cotguty class Cotguty class for rego (frst-order spatal lag: (.a L j j j Cotguty class comprses all mmedate eghbours of a rego (= frstorder eghbours Frst-order spatal lag for all regos: (.b L W Attrbute vector (observatos of a geo-refereced var.: = ( W: (Stadardzed cotguty matr (= spatal eghts matr Cotguty class (secod-order spatal lag: (.a L j ( j j (.b W Cotguty class comprses all secod-order eghbours ( j W : Secod-order cotguty matr, : elemets of W L 5

6 Cotguty class k (k-th order spatal lag: (.a (.b L k L k (k j j W, k j, k,,... k,,... Cotguty class k comprses all k-th order eghbours W k : k-th order cotguty matr, (k j : elemets of W k Spatal lag ad dstace-based spatal eght matr: Istead of the cotguty matr, a spatal lag ca be formed for a dstace-based spatal eght W. I ths case oly a frst-order lag s accessble to terpretato. The spatal lag s the alays gve by (.a ad (.b, respectvely. Spatal lag ad spatal autocorrelato: Measures of spatal autocorrelato make use of the cocept of the spatal lag. For a quattatve geo-refereced varable X, spatal autocorrelato ca be assessed by calbratg ts observato vector ad the spatal lag W. I order to preserve the propertes of correlato coeffcets, the stadardzed spatal eght matr W s geerally preferred to the ustadardzed eght matr W*. 6

7 . Global spatal autocorrelato.. Mora s I Mora's I s the mostly used measure of global spatal autocorrelato. It ca be appled to detect departures from spatal radomess. Departures from radomess dcate spatal patters such as clusters or treds over space. Mora s I s based o cross-products to measure spatal autocorrelato. It measures the degree of lear assocato betee a the vector of observed values of a geo-refereced varable X ad ts spatal lag L,.e. a eghted average of the eghbourg values. Mora s I th ustadardzed spatal eght matr W*: (.4a th (.5 I S S * j j * j ( ( j (umber of cross-products ( ( j j S ( ( Aalogy to the covetal correlato coeffcet: Numerator: sum of cross-products, Deomator: sum of squared devatos * j j 7

8 I matr otato: (.4b I S ( ' W*( ( '( : vector of observatos of X : vector cotag the mea of X Mora s I th stadardzed spatal eght matr W: j ( ( j ( j j j (.6a I ( ( ( j I matr otato: (.6b ( ' W( I ( '( Rage of Mora s I case of stadardzed eght matr: - I (ot garateed for ustadardzed eght matr 8

9 Mora s I ad lear regresso: Formally I s equvalet to the slope coeffcet of a lear regresso of the spatal lag W o the observato vector measured devatos from ther meas. It s, hoever, ot equvalet to the slope of a lear regresso of o W hch ould be a more atural ay to specfy a spatal process. A specal form of ths scatterplot ( Secto..: Mora scatterplot ca be used to assess the degree of ft, detfy outlers ad leverage pots as ell as local pockets of statoarty. 9

10 Spatal patter of a geo-refereced varable Radom patter Mora s I = -. Spatal clusterg Mora s I =.486

11 Eample: Fgure: Arragemet of spatal uts 4 5 Cotguty matr: Geo-refereced varable X: Uemploymet ( % * W 55 Rego Uempoymet rate

12 A. Calculato of Mora s I th ustadardzed eght matr Calculato th sum of cross-products ad sum of squares * j ( ( j j I S ( Arthmetc mea (=5: ( Table : Cross-products ( ( j Rego 4 5 (8-5 = =9 (8-5(6-5= (8-5(6-5= (8-5(-5=-6 (8-5(-5=-9 (6-5(8-5= (6-5 = = (6-5(6-5= (6-5(-5=- (6-5(-5=- (6-5(8-5= (6-5(6-5= (6-5 = = (6-5(-5=- (6-5(-5=- 4 (-5(8-5=-6 (-5(6-5=- (-5(6-5=- (-5 =(- =4 (-5(-5=6 5 (-5(8-5=-9 (-5(6-5=- (-5(6-5=- (-5(-5=6 (-5 =(- =9

13 Table : Ustadardzed eghts * j Rego Sum of eghts (# of o-zero eghts S = * Table : Weghted cross-products j ( ( j Rego = = = (-6 = (-9 = = = = (- = - (- = = = = (- = - (- = 4 (-6 = (- = - (- = - 4 = 6 = 6 5 (-9 = (- = (- = 6 = 6 9 = Sum of eghted cross-products 8

14 Table 4: Sum of squared devatos ( Rego = 9 ( = = 4 5 = = - 9 Sum of squared devatos 4 Mora s I (ustadardzed eght matr: ( = I

15 5 Compact calculato th observato vector ad eght matr Numerator (quadratc form: '( ( *( ' ( S I W (th = 5 *( ' ( W = 8 '

16 Deomator (scalar product: ( ' '( = 4 Mora s I (ustadardzed eght matr: ( = 5, S = 5 8 I

17 A. Calculato of Mora s I th stadardzed eght matr Calculato th sum of cross-products ad sum of squares j ( ( j j I ( Arthmetc mea (=5: ( Table 5: Cross-products ( ( j Rego 4 5 (8-5 = =9 (8-5(6-5= (8-5(6-5= (8-5(-5=-6 (8-5(-5=-9 (6-5(8-5= (6-5 = = (6-5(6-5= (6-5(-5=- (6-5(-5=- (6-5(8-5= (6-5(6-5= (6-5 = = (6-5(-5=- (6-5(-5=- 4 (-5(8-5=-6 (-5(6-5=- (-5(6-5=- (-5 =(- =4 (-5(-5=6 5 (-5(8-5=-9 (-5(6-5=- (-5(6-5=- (-5(-5=6 (-5 =(- =9 7

18 Table 6: Stadardzed eghts j Rego 4 5 / / / / / / / / 4 / / / 5 Table 7: Weghted cross-products j ( ( j Rego = (½ =,5 (½ =,5 (-6 = (-9 = (/ = = (/ = / (/(- = -/ (- = (/ = (/ = / = (/(- = -/ (- = 4 (-6 = (/(- = -/ (/(- = -/ 4 = (/6 = 5 (-9 = (- = (- = 6 = 6 9 = Sum of eghted cross-products 8

19 Table 8: Sum of squared devatos ( Rego = 9 ( = = 4 5 = = - 9 Sum of squared devatos 4 Mora s I (stadardzed eght matr: ( = 5 I

20 Compact calculato th observato vector ad eght matr Numerator (quadratc form: (th = 5 ( ' ( W / / / = ' '( ( ( ' ( I W / / / / / / / / / / /

21 Deomator (scalar product: ( ' '( = 4 Mora s I (stadardzed eght matr: ( = 5 I

22 Sgfcace test of Mora s I Sgfcace of Mora s I ca be assessed uder ormal appromato or radomzato. The varace formula for ormal appromato s much smpler tha for radomzato. Here e preset the sgfcace test of Mora s I for the ormal appromato by makg use of a ro-stadardsed eghts matr. Null hypothess H : No spatal autocorrelato (spatal radomess Alteratve hypothess H : Spatal autocorrelato (spatal depedece or Alteratve hypothess H : Postve Spatal autocorrelato Test statstc: (.7 Epected value: (.8 I E(I Z(I Var(I a E(I ~ N(, Var(I Varace (for ormal appro.: (.9,, S ( S S S E(I

23 S j j th S (j j j S j j ( j j th j j Test decso (rght-sded test: z(i > z -α => reject H (postve spatal autocorrelato z(i: z-score, α: sgfcace level, z -α : (-α-quatle of stad. ormal dstrbuto

24 Eample: I the eample of 5 regos (=5 e have calculated a value of Mora s I of,458 o the bass of the stadardzed eght matr: / W / Despte the small sample e use the ormal appromato of the sgfcace test of Mora s I for llustratve purposes. / / / Epected value: E(I, 5 4 / / / / / / S j j 5 4

25 / ( 5/ ( 6 7 / ( 6 7 / ( / ( ( ( ( ( ( ( S j j j j ]/ [ ]/ / ( (/ / (/ / (/ / (/ / (/ / (/ / (/ / (/ / (/ / (/ / (/ ( ( ( ( ( ( ( ( ( ( ( [( ( S j j j

26 Var(I 5 S ( S S (5 5 S,7,65,745 4 E(I Test statstc (z-score: z(i I E(I Var(I Crtcal value (α=,5, rght-sded test: z.95 =.6449 Test decso: z(i =.5955 > z.95 =.6449 => Reject H Iterpretato: Sgfcat postve spatal autocorrelato of the uemploymet rate 6

27 .. Mora scatterplot The terpretato of Mora s I as the slope of a regresso le provdes a ay to vsualze the lear assocato form of a bvarate scatterplot of W agast he both vectors are measured devatos from ther meas. The Mora scatterplot s a specal form of a bvarate scatterplot hch makes use of the stadardzed values of the pars (, L. Augmeted th the regresso le, t ca be used to assess the degree of ft ad to detfy outlers ad leverage pots. Moreover, the Mora scatterplot ca be used to detfy local pockets of statoarty. Table: Types of local spatal assocato Geo-refereced varable (X Spatally lagged geo-refereced varable (LX hgh lo hgh Quadrat I: HH Quadrat IV: HL lo Quadrat II: LH Quadrat III: LL Postve spatal assocato: Quadrats I (HH ad III (LL Negatve spatal assocato: Quadrats II (LH ad IV (HL Spatal outlers case of postve global spatal autocorrelato 7

28 Leverage pots: Outlers the eplaatory varable hch have the potetal to affect the posto of the regresso le. Leverage pots ca but do ot ecessarly dstort the regresso coeffcets. Idetfcato of leverage pots: Dagoal elemets h of the hat matr (. H X( X' X X' (hat matr: for th observato: ' h ( X' X Geeral propertes of the hat matr H: H X( X' X X' Symmetry: H = H ad I - H = (I H Idempotece: HH = H ad (I H(I H = I H Ftted values (th projecto matr: Resdual maker : e = (I H y Varace-covarace matr of : Etreme leverage: h > 4/ (rule of thumb e yˆ Hy Var( e ( I H 8

29 Ifluetal observatos: Observatos that eert a large fluece o the regresso le. Ifluetal observatos ca be deoted as harmful leverage pots. Cook s dstace: (. D e h k s ( h k: umber of eplaatory varables (c. tercept s : Ubased estmate of the error varace Cut-off value: D > (strog fluetal observato s e /( k D > 4/(-k (otable fluetal observato; Fo, 99 Geeral outlers (to-sgma rule: Pots further tha stadard devatos aay from the org 9

30 Eample: I the eample of 5 regos the uemploymet rate s cosdered as the georefereced varable ( %. The observato vector s gve by We calculate the spatal lag L of the uemploymet rate usg the stadardzed eght matr W: / / 6 (6 / 6 (8 / 6 (8 6 / ( / / / / / / / / / / / L W

31 Regresso: L o Regresso le: ^ L a b Ordary least squares (OLS estmators for a ad b: (. bˆ (. ( Workg table: L L â L bˆ L L / / / Regresso le: L ^ OLS estmators: bˆ (slope = Mora s I â 5 5,

32 ^ Regresso values L : ^ L ^ L ^ L ^ L 4 ^,785,4588 6,749,785,4586 5,458,785,4586 5,458,785,458 4,84 L 5,785,458,65 ^ Ftted values L ad resduals e : L ^ L e 8 6 6,749 -, / 5,458, / 5,458, / 4,84,58 5,65 -,

33 Mora Scatterplot Stadardzato of the uemploymet rate ad t lagged values 5 5 s. Table: s ,8 5 4,8 s 4,8 5 5 s. Table:49 Stadardzed values (z-values of the -values: z s,9 z Lagged values z : Lz z = (8-5 /.9 =.69 Lz = =.456 z = (6-5 /.9 =.456 Lz = (/.69 + (/ (/ (-.9 =.4 z = (6-5 /.9 =.456 Lz = (/.69 + (/ (/ (-.9 =.4 z 4 = ( - 5 /.9 = -.9 Lz 4 = (/ (/ (/ (-.69 = -.5 z 5 = ( - 5 /.9 = -.69 Lz 5 = (-.9 = -.9

34 Regresso le for the stadardzed values (Mora scatterplot - Loest z value (rego 5: z 5 = -.69 L 5.65 L 5 Stadardzed L ẑ 5 L5 s value: Largest z value (rego : z =.69 L 6.75 Stadardzed L ẑ L s L value:

35 Lz Mora scatterplot (5 regos Mora scatterplot (5 regos z 5

36 R-Code for Mora Scatterplot # Mora scatterplot (eample scrpt = c(8, 6, 6,, = legth( WS = matr(c(,,,,,,,,,,,,,,,,,,,,,,,,, ro=, col= s.rsum = apply(ws,, sum W = WS/s.rsum L = W%*%.lm = lm(l~.lm L.ft = L.lm$ftted.values m = mea( s = sqrt((-/*sd( z = (-m/s Lz = W%*%z lz.ft = (L.ft - m/s plot(z, Lz, ma="mora scatterplot (5 regos",lm=c(-.5,.5, ylm=c(-,, ce.ma=.5, fot.ma= able(h=, v= les(z, Lz.ft bo(hch="fgure" 6

37 Table: Types of local spatal assocato Uemploymet rate (X hgh lo Spatally lagged uemploymet rate (LX hgh Quadrat I: HH Regos, ad Quadrat II: LH - lo Quadrat IV: HL - Quadrat III: LL Regos 4 ad 5 Postve spatal assocato: all regos (spatal clusters Negatve spatal assocato: o rego (o spatal outlers Geeral outler detecto: No observatos outsde the [-; ] terval No outlers accordg ot the to-sgma rule 7

38 8 Hat matr ad leverage pots X Observato matr: X'X,47,8,8,47 ' ( X X

39 H X ( X ' X,45,8,8,667,85,575,5,5,5,75 X ',5,47,47 8,8,5,5,47,47,67,75 8 6,47 6,8 6,5,47,47,67,75 6,8,47 8,5,67,67,667,45 6,75,75,75,45,575 Dagoal elemets of the hat matr H (cut-off value: 4/(=5=,8: h =,575, h =,47, h =,47, h 4 =,667, h 5 =,

40 Cook s dstace s : Ubased estmate of the error varace s e /( k,9584/(5,95 e h,46,575,8845 k s ( h,95 (,575,549 e h,44,47,49 k s ( h,95 (,47,6747 D D D D D e h,44,47,49 k s ( h,95 (,47,6747 e4 h4,4,667,475 4 k s ( h,95 (,667,568 4 e5 h5,98,575,47 5 k s ( h,95 (,575,549 5,74,85,85,4868,9469 Rego 5: (D 5 =,9469 > (=cut-off value => Ifluetal observato 4

Simple Linear Regression

Simple Linear Regression Statstcal Methods I (EST 75) Page 139 Smple Lear Regresso Smple regresso applcatos are used to ft a model descrbg a lear relatoshp betwee two varables. The aspects of least squares regresso ad correlato

More information

4. Standard Regression Model and Spatial Dependence Tests

4. Standard Regression Model and Spatial Dependence Tests 4. Stadard Regresso Model ad Spatal Depedece Tests Stadard regresso aalss fals the presece of spatal effects. I case of spatal depedeces ad/or spatal heterogeet a stadard regresso model wll be msspecfed.

More information

Lecture 7. Confidence Intervals and Hypothesis Tests in the Simple CLR Model

Lecture 7. Confidence Intervals and Hypothesis Tests in the Simple CLR Model Lecture 7. Cofdece Itervals ad Hypothess Tests the Smple CLR Model I lecture 6 we troduced the Classcal Lear Regresso (CLR) model that s the radom expermet of whch the data Y,,, K, are the outcomes. The

More information

Chapter 13 Student Lecture Notes 13-1

Chapter 13 Student Lecture Notes 13-1 Chapter 3 Studet Lecture Notes 3- Basc Busess Statstcs (9 th Edto) Chapter 3 Smple Lear Regresso 4 Pretce-Hall, Ic. Chap 3- Chapter Topcs Types of Regresso Models Determg the Smple Lear Regresso Equato

More information

Econometric Methods. Review of Estimation

Econometric Methods. Review of Estimation Ecoometrc Methods Revew of Estmato Estmatg the populato mea Radom samplg Pot ad terval estmators Lear estmators Ubased estmators Lear Ubased Estmators (LUEs) Effcecy (mmum varace) ad Best Lear Ubased Estmators

More information

Simple Linear Regression

Simple Linear Regression Correlato ad Smple Lear Regresso Berl Che Departmet of Computer Scece & Iformato Egeerg Natoal Tawa Normal Uversty Referece:. W. Navd. Statstcs for Egeerg ad Scetsts. Chapter 7 (7.-7.3) & Teachg Materal

More information

Summary of the lecture in Biostatistics

Summary of the lecture in Biostatistics Summary of the lecture Bostatstcs Probablty Desty Fucto For a cotuos radom varable, a probablty desty fucto s a fucto such that: 0 dx a b) b a dx A probablty desty fucto provdes a smple descrpto of the

More information

Lecture 3 Probability review (cont d)

Lecture 3 Probability review (cont d) STATS 00: Itroducto to Statstcal Iferece Autum 06 Lecture 3 Probablty revew (cot d) 3. Jot dstrbutos If radom varables X,..., X k are depedet, the ther dstrbuto may be specfed by specfyg the dvdual dstrbuto

More information

Probability and. Lecture 13: and Correlation

Probability and. Lecture 13: and Correlation 933 Probablty ad Statstcs for Software ad Kowledge Egeers Lecture 3: Smple Lear Regresso ad Correlato Mocha Soptkamo, Ph.D. Outle The Smple Lear Regresso Model (.) Fttg the Regresso Le (.) The Aalyss of

More information

ENGI 3423 Simple Linear Regression Page 12-01

ENGI 3423 Simple Linear Regression Page 12-01 ENGI 343 mple Lear Regresso Page - mple Lear Regresso ometmes a expermet s set up where the expermeter has cotrol over the values of oe or more varables X ad measures the resultg values of aother varable

More information

b. There appears to be a positive relationship between X and Y; that is, as X increases, so does Y.

b. There appears to be a positive relationship between X and Y; that is, as X increases, so does Y. .46. a. The frst varable (X) s the frst umber the par ad s plotted o the horzotal axs, whle the secod varable (Y) s the secod umber the par ad s plotted o the vertcal axs. The scatterplot s show the fgure

More information

Objectives of Multiple Regression

Objectives of Multiple Regression Obectves of Multple Regresso Establsh the lear equato that best predcts values of a depedet varable Y usg more tha oe eplaator varable from a large set of potetal predctors {,,... k }. Fd that subset of

More information

Mean is only appropriate for interval or ratio scales, not ordinal or nominal.

Mean is only appropriate for interval or ratio scales, not ordinal or nominal. Mea Same as ordary average Sum all the data values ad dvde by the sample sze. x = ( x + x +... + x Usg summato otato, we wrte ths as x = x = x = = ) x Mea s oly approprate for terval or rato scales, ot

More information

Ordinary Least Squares Regression. Simple Regression. Algebra and Assumptions.

Ordinary Least Squares Regression. Simple Regression. Algebra and Assumptions. Ordary Least Squares egresso. Smple egresso. Algebra ad Assumptos. I ths part of the course we are gog to study a techque for aalysg the lear relatoshp betwee two varables Y ad X. We have pars of observatos

More information

Chapter 14 Logistic Regression Models

Chapter 14 Logistic Regression Models Chapter 4 Logstc Regresso Models I the lear regresso model X β + ε, there are two types of varables explaatory varables X, X,, X k ad study varable y These varables ca be measured o a cotuous scale as

More information

Correlation and Simple Linear Regression

Correlation and Simple Linear Regression Correlato ad Smple Lear Regresso Berl Che Departmet of Computer Scece & Iformato Egeerg Natoal Tawa Normal Uverst Referece:. W. Navd. Statstcs for Egeerg ad Scetsts. Chapter 7 (7.-7.3) & Teachg Materal

More information

STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. x, where. = y - ˆ " 1

STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. x, where. = y - ˆ  1 STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS Recall Assumpto E(Y x) η 0 + η x (lear codtoal mea fucto) Data (x, y ), (x 2, y 2 ),, (x, y ) Least squares estmator ˆ E (Y x) ˆ " 0 + ˆ " x, where ˆ

More information

ECONOMETRIC THEORY. MODULE VIII Lecture - 26 Heteroskedasticity

ECONOMETRIC THEORY. MODULE VIII Lecture - 26 Heteroskedasticity ECONOMETRIC THEORY MODULE VIII Lecture - 6 Heteroskedastcty Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur . Breusch Paga test Ths test ca be appled whe the replcated data

More information

Statistics MINITAB - Lab 5

Statistics MINITAB - Lab 5 Statstcs 10010 MINITAB - Lab 5 PART I: The Correlato Coeffcet Qute ofte statstcs we are preseted wth data that suggests that a lear relatoshp exsts betwee two varables. For example the plot below s of

More information

ESS Line Fitting

ESS Line Fitting ESS 5 014 17. Le Fttg A very commo problem data aalyss s lookg for relatoshpetwee dfferet parameters ad fttg les or surfaces to data. The smplest example s fttg a straght le ad we wll dscuss that here

More information

Linear Regression with One Regressor

Linear Regression with One Regressor Lear Regresso wth Oe Regressor AIM QA.7. Expla how regresso aalyss ecoometrcs measures the relatoshp betwee depedet ad depedet varables. A regresso aalyss has the goal of measurg how chages oe varable,

More information

Lecture Notes Types of economic variables

Lecture Notes Types of economic variables Lecture Notes 3 1. Types of ecoomc varables () Cotuous varable takes o a cotuum the sample space, such as all pots o a le or all real umbers Example: GDP, Polluto cocetrato, etc. () Dscrete varables fte

More information

COV. Violation of constant variance of ε i s but they are still independent. The error term (ε) is said to be heteroscedastic.

COV. Violation of constant variance of ε i s but they are still independent. The error term (ε) is said to be heteroscedastic. c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty olato of costat varace of s but they are stll depedet. C,, he error term s sad to be heteroscedastc. c Pogsa Porchawseskul, Faculty of Ecoomcs,

More information

Statistics. Correlational. Dr. Ayman Eldeib. Simple Linear Regression and Correlation. SBE 304: Linear Regression & Correlation 1/3/2018

Statistics. Correlational. Dr. Ayman Eldeib. Simple Linear Regression and Correlation. SBE 304: Linear Regression & Correlation 1/3/2018 /3/08 Sstems & Bomedcal Egeerg Departmet SBE 304: Bo-Statstcs Smple Lear Regresso ad Correlato Dr. Ama Eldeb Fall 07 Descrptve Orgasg, summarsg & descrbg data Statstcs Correlatoal Relatoshps Iferetal Geeralsg

More information

Chapter Business Statistics: A First Course Fifth Edition. Learning Objectives. Correlation vs. Regression. In this chapter, you learn:

Chapter Business Statistics: A First Course Fifth Edition. Learning Objectives. Correlation vs. Regression. In this chapter, you learn: Chapter 3 3- Busess Statstcs: A Frst Course Ffth Edto Chapter 2 Correlato ad Smple Lear Regresso Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic. Chap 2- Learg Objectves I ths chapter, you lear:

More information

Multiple Linear Regression Analysis

Multiple Linear Regression Analysis LINEA EGESSION ANALYSIS MODULE III Lecture - 4 Multple Lear egresso Aalyss Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur Cofdece terval estmato The cofdece tervals multple

More information

Multiple Regression. More than 2 variables! Grade on Final. Multiple Regression 11/21/2012. Exam 2 Grades. Exam 2 Re-grades

Multiple Regression. More than 2 variables! Grade on Final. Multiple Regression 11/21/2012. Exam 2 Grades. Exam 2 Re-grades STAT 101 Dr. Kar Lock Morga 11/20/12 Exam 2 Grades Multple Regresso SECTIONS 9.2, 10.1, 10.2 Multple explaatory varables (10.1) Parttog varablty R 2, ANOVA (9.2) Codtos resdual plot (10.2) Trasformatos

More information

To use adaptive cluster sampling we must first make some definitions of the sampling universe:

To use adaptive cluster sampling we must first make some definitions of the sampling universe: 8.3 ADAPTIVE SAMPLING Most of the methods dscussed samplg theory are lmted to samplg desgs hch the selecto of the samples ca be doe before the survey, so that oe of the decsos about samplg deped ay ay

More information

STA302/1001-Fall 2008 Midterm Test October 21, 2008

STA302/1001-Fall 2008 Midterm Test October 21, 2008 STA3/-Fall 8 Mdterm Test October, 8 Last Name: Frst Name: Studet Number: Erolled (Crcle oe) STA3 STA INSTRUCTIONS Tme allowed: hour 45 mutes Ads allowed: A o-programmable calculator A table of values from

More information

Example: Multiple linear regression. Least squares regression. Repetition: Simple linear regression. Tron Anders Moger

Example: Multiple linear regression. Least squares regression. Repetition: Simple linear regression. Tron Anders Moger Example: Multple lear regresso 5000,00 4000,00 Tro Aders Moger 0.0.007 brthweght 3000,00 000,00 000,00 0,00 50,00 00,00 50,00 00,00 50,00 weght pouds Repetto: Smple lear regresso We defe a model Y = β0

More information

hp calculators HP 30S Statistics Averages and Standard Deviations Average and Standard Deviation Practice Finding Averages and Standard Deviations

hp calculators HP 30S Statistics Averages and Standard Deviations Average and Standard Deviation Practice Finding Averages and Standard Deviations HP 30S Statstcs Averages ad Stadard Devatos Average ad Stadard Devato Practce Fdg Averages ad Stadard Devatos HP 30S Statstcs Averages ad Stadard Devatos Average ad stadard devato The HP 30S provdes several

More information

Application of Calibration Approach for Regression Coefficient Estimation under Two-stage Sampling Design

Application of Calibration Approach for Regression Coefficient Estimation under Two-stage Sampling Design Authors: Pradp Basak, Kaustav Adtya, Hukum Chadra ad U.C. Sud Applcato of Calbrato Approach for Regresso Coeffcet Estmato uder Two-stage Samplg Desg Pradp Basak, Kaustav Adtya, Hukum Chadra ad U.C. Sud

More information

Module 7: Probability and Statistics

Module 7: Probability and Statistics Lecture 4: Goodess of ft tests. Itroducto Module 7: Probablty ad Statstcs I the prevous two lectures, the cocepts, steps ad applcatos of Hypotheses testg were dscussed. Hypotheses testg may be used to

More information

TESTS BASED ON MAXIMUM LIKELIHOOD

TESTS BASED ON MAXIMUM LIKELIHOOD ESE 5 Toy E. Smth. The Basc Example. TESTS BASED ON MAXIMUM LIKELIHOOD To llustrate the propertes of maxmum lkelhood estmates ad tests, we cosder the smplest possble case of estmatg the mea of the ormal

More information

Outline. Point Pattern Analysis Part I. Revisit IRP/CSR

Outline. Point Pattern Analysis Part I. Revisit IRP/CSR Pot Patter Aalyss Part I Outle Revst IRP/CSR, frst- ad secod order effects What s pot patter aalyss (PPA)? Desty-based pot patter measures Dstace-based pot patter measures Revst IRP/CSR Equal probablty:

More information

Regresso What s a Model? 1. Ofte Descrbe Relatoshp betwee Varables 2. Types - Determstc Models (o radomess) - Probablstc Models (wth radomess) EPI 809/Sprg 2008 9 Determstc Models 1. Hypothesze

More information

Module 7. Lecture 7: Statistical parameter estimation

Module 7. Lecture 7: Statistical parameter estimation Lecture 7: Statstcal parameter estmato Parameter Estmato Methods of Parameter Estmato 1) Method of Matchg Pots ) Method of Momets 3) Mamum Lkelhood method Populato Parameter Sample Parameter Ubased estmato

More information

12.2 Estimating Model parameters Assumptions: ox and y are related according to the simple linear regression model

12.2 Estimating Model parameters Assumptions: ox and y are related according to the simple linear regression model 1. Estmatg Model parameters Assumptos: ox ad y are related accordg to the smple lear regresso model (The lear regresso model s the model that says that x ad y are related a lear fasho, but the observed

More information

Recall MLR 5 Homskedasticity error u has the same variance given any values of the explanatory variables Var(u x1,...,xk) = 2 or E(UU ) = 2 I

Recall MLR 5 Homskedasticity error u has the same variance given any values of the explanatory variables Var(u x1,...,xk) = 2 or E(UU ) = 2 I Chapter 8 Heterosedastcty Recall MLR 5 Homsedastcty error u has the same varace gve ay values of the eplaatory varables Varu,..., = or EUU = I Suppose other GM assumptos hold but have heterosedastcty.

More information

Simple Linear Regression and Correlation.

Simple Linear Regression and Correlation. Smple Lear Regresso ad Correlato. Correspods to Chapter 0 Tamhae ad Dulop Sldes prepared b Elzabeth Newto (MIT) wth some sldes b Jacquele Telford (Johs Hopks Uverst) Smple lear regresso aalss estmates

More information

Simple Linear Regression - Scalar Form

Simple Linear Regression - Scalar Form Smple Lear Regresso - Scalar Form Q.. Model Y X,..., p..a. Derve the ormal equatos that mmze Q. p..b. Solve for the ordary least squares estmators, p..c. Derve E, V, E, V, COV, p..d. Derve the mea ad varace

More information

is the score of the 1 st student, x

is the score of the 1 st student, x 8 Chapter Collectg, Dsplayg, ad Aalyzg your Data. Descrptve Statstcs Sectos explaed how to choose a sample, how to collect ad orgaze data from the sample, ad how to dsplay your data. I ths secto, you wll

More information

( ) = ( ) ( ) Chapter 13 Asymptotic Theory and Stochastic Regressors. Stochastic regressors model

( ) = ( ) ( ) Chapter 13 Asymptotic Theory and Stochastic Regressors. Stochastic regressors model Chapter 3 Asmptotc Theor ad Stochastc Regressors The ature of eplaator varable s assumed to be o-stochastc or fed repeated samples a regresso aalss Such a assumpto s approprate for those epermets whch

More information

Midterm Exam 1, section 1 (Solution) Thursday, February hour, 15 minutes

Midterm Exam 1, section 1 (Solution) Thursday, February hour, 15 minutes coometrcs, CON Sa Fracsco State Uversty Mchael Bar Sprg 5 Mdterm am, secto Soluto Thursday, February 6 hour, 5 mutes Name: Istructos. Ths s closed book, closed otes eam.. No calculators of ay kd are allowed..

More information

best estimate (mean) for X uncertainty or error in the measurement (systematic, random or statistical) best

best estimate (mean) for X uncertainty or error in the measurement (systematic, random or statistical) best Error Aalyss Preamble Wheever a measuremet s made, the result followg from that measuremet s always subject to ucertaty The ucertaty ca be reduced by makg several measuremets of the same quatty or by mprovg

More information

ENGI 4421 Joint Probability Distributions Page Joint Probability Distributions [Navidi sections 2.5 and 2.6; Devore sections

ENGI 4421 Joint Probability Distributions Page Joint Probability Distributions [Navidi sections 2.5 and 2.6; Devore sections ENGI 441 Jot Probablty Dstrbutos Page 7-01 Jot Probablty Dstrbutos [Navd sectos.5 ad.6; Devore sectos 5.1-5.] The jot probablty mass fucto of two dscrete radom quattes, s, P ad p x y x y The margal probablty

More information

residual. (Note that usually in descriptions of regression analysis, upper-case

residual. (Note that usually in descriptions of regression analysis, upper-case Regresso Aalyss Regresso aalyss fts or derves a model that descres the varato of a respose (or depedet ) varale as a fucto of oe or more predctor (or depedet ) varales. The geeral regresso model s oe of

More information

IFYMB002 Mathematics Business Appendix C Formula Booklet

IFYMB002 Mathematics Business Appendix C Formula Booklet Iteratoal Foudato Year (IFY IFYMB00 Mathematcs Busess Apped C Formula Booklet Related Documet: IFY Mathematcs Busess Syllabus 07/8 IFYMB00 Maths Busess Apped C Formula Booklet Cotets lease ote that the

More information

Wu-Hausman Test: But if X and ε are independent, βˆ. ECON 324 Page 1

Wu-Hausman Test: But if X and ε are independent, βˆ. ECON 324 Page 1 Wu-Hausma Test: Detectg Falure of E( ε X ) Caot drectly test ths assumpto because lack ubased estmator of ε ad the OLS resduals wll be orthogoal to X, by costructo as ca be see from the momet codto X'

More information

THE ROYAL STATISTICAL SOCIETY HIGHER CERTIFICATE

THE ROYAL STATISTICAL SOCIETY HIGHER CERTIFICATE THE ROYAL STATISTICAL SOCIETY 00 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE PAPER I STATISTICAL THEORY The Socety provdes these solutos to assst caddates preparg for the examatos future years ad for the

More information

Chapter 9 Jordan Block Matrices

Chapter 9 Jordan Block Matrices Chapter 9 Jorda Block atrces I ths chapter we wll solve the followg problem. Gve a lear operator T fd a bass R of F such that the matrx R (T) s as smple as possble. f course smple s a matter of taste.

More information

Multiple Choice Test. Chapter Adequacy of Models for Regression

Multiple Choice Test. Chapter Adequacy of Models for Regression Multple Choce Test Chapter 06.0 Adequac of Models for Regresso. For a lear regresso model to be cosdered adequate, the percetage of scaled resduals that eed to be the rage [-,] s greater tha or equal to

More information

Simple Linear Regression and Correlation. Applied Statistics and Probability for Engineers. Chapter 11 Simple Linear Regression and Correlation

Simple Linear Regression and Correlation. Applied Statistics and Probability for Engineers. Chapter 11 Simple Linear Regression and Correlation 4//6 Appled Statstcs ad Probablty for Egeers Sth Edto Douglas C. Motgomery George C. Ruger Chapter Smple Lear Regresso ad Correlato CHAPTER OUTLINE Smple Lear Regresso ad Correlato - Emprcal Models -8

More information

Functions of Random Variables

Functions of Random Variables Fuctos of Radom Varables Chapter Fve Fuctos of Radom Varables 5. Itroducto A geeral egeerg aalyss model s show Fg. 5.. The model output (respose) cotas the performaces of a system or product, such as weght,

More information

Fundamentals of Regression Analysis

Fundamentals of Regression Analysis Fdametals of Regresso Aalyss Regresso aalyss s cocered wth the stdy of the depedece of oe varable, the depedet varable, o oe or more other varables, the explaatory varables, wth a vew of estmatg ad/or

More information

ECON 482 / WH Hong The Simple Regression Model 1. Definition of the Simple Regression Model

ECON 482 / WH Hong The Simple Regression Model 1. Definition of the Simple Regression Model ECON 48 / WH Hog The Smple Regresso Model. Defto of the Smple Regresso Model Smple Regresso Model Expla varable y terms of varable x y = β + β x+ u y : depedet varable, explaed varable, respose varable,

More information

CS 2750 Machine Learning. Lecture 8. Linear regression. CS 2750 Machine Learning. Linear regression. is a linear combination of input components x

CS 2750 Machine Learning. Lecture 8. Linear regression. CS 2750 Machine Learning. Linear regression. is a linear combination of input components x CS 75 Mache Learg Lecture 8 Lear regresso Mlos Hauskrecht mlos@cs.ptt.edu 539 Seott Square CS 75 Mache Learg Lear regresso Fucto f : X Y s a lear combato of put compoets f + + + K d d K k - parameters

More information

Chapter 8. Inferences about More Than Two Population Central Values

Chapter 8. Inferences about More Than Two Population Central Values Chapter 8. Ifereces about More Tha Two Populato Cetral Values Case tudy: Effect of Tmg of the Treatmet of Port-We tas wth Lasers ) To vestgate whether treatmet at a youg age would yeld better results tha

More information

Chapter 5 Properties of a Random Sample

Chapter 5 Properties of a Random Sample Lecture 6 o BST 63: Statstcal Theory I Ku Zhag, /0/008 Revew for the prevous lecture Cocepts: t-dstrbuto, F-dstrbuto Theorems: Dstrbutos of sample mea ad sample varace, relatoshp betwee sample mea ad sample

More information

Lecture 1 Review of Fundamental Statistical Concepts

Lecture 1 Review of Fundamental Statistical Concepts Lecture Revew of Fudametal Statstcal Cocepts Measures of Cetral Tedecy ad Dsperso A word about otato for ths class: Idvduals a populato are desgated, where the dex rages from to N, ad N s the total umber

More information

X X X E[ ] E X E X. is the ()m n where the ( i,)th. j element is the mean of the ( i,)th., then

X X X E[ ] E X E X. is the ()m n where the ( i,)th. j element is the mean of the ( i,)th., then Secto 5 Vectors of Radom Varables Whe workg wth several radom varables,,..., to arrage them vector form x, t s ofte coveet We ca the make use of matrx algebra to help us orgaze ad mapulate large umbers

More information

Midterm Exam 1, section 2 (Solution) Thursday, February hour, 15 minutes

Midterm Exam 1, section 2 (Solution) Thursday, February hour, 15 minutes coometrcs, CON Sa Fracsco State Uverst Mchael Bar Sprg 5 Mdterm xam, secto Soluto Thursda, Februar 6 hour, 5 mutes Name: Istructos. Ths s closed book, closed otes exam.. No calculators of a kd are allowed..

More information

Arithmetic Mean Suppose there is only a finite number N of items in the system of interest. Then the population arithmetic mean is

Arithmetic Mean Suppose there is only a finite number N of items in the system of interest. Then the population arithmetic mean is Topc : Probablty Theory Module : Descrptve Statstcs Measures of Locato Descrptve statstcs are measures of locato ad shape that perta to probablty dstrbutos The prmary measures of locato are the arthmetc

More information

MEASURES OF DISPERSION

MEASURES OF DISPERSION MEASURES OF DISPERSION Measure of Cetral Tedecy: Measures of Cetral Tedecy ad Dsperso ) Mathematcal Average: a) Arthmetc mea (A.M.) b) Geometrc mea (G.M.) c) Harmoc mea (H.M.) ) Averages of Posto: a) Meda

More information

Measures of Dispersion

Measures of Dispersion Chapter 8 Measures of Dsperso Defto of Measures of Dsperso (page 31) A measure of dsperso s a descrptve summary measure that helps us characterze the data set terms of how vared the observatos are from

More information

Special Instructions / Useful Data

Special Instructions / Useful Data JAM 6 Set of all real umbers P A..d. B, p Posso Specal Istructos / Useful Data x,, :,,, x x Probablty of a evet A Idepedetly ad detcally dstrbuted Bomal dstrbuto wth parameters ad p Posso dstrbuto wth

More information

Generative classification models

Generative classification models CS 75 Mache Learg Lecture Geeratve classfcato models Mlos Hauskrecht mlos@cs.ptt.edu 539 Seott Square Data: D { d, d,.., d} d, Classfcato represets a dscrete class value Goal: lear f : X Y Bar classfcato

More information

THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA

THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA THE ROYAL STATISTICAL SOCIETY EXAMINATIONS SOLUTIONS GRADUATE DIPLOMA PAPER II STATISTICAL THEORY & METHODS The Socety provdes these solutos to assst caddates preparg for the examatos future years ad for

More information

Chapter 13, Part A Analysis of Variance and Experimental Design. Introduction to Analysis of Variance. Introduction to Analysis of Variance

Chapter 13, Part A Analysis of Variance and Experimental Design. Introduction to Analysis of Variance. Introduction to Analysis of Variance Chapter, Part A Aalyss of Varace ad Epermetal Desg Itroducto to Aalyss of Varace Aalyss of Varace: Testg for the Equalty of Populato Meas Multple Comparso Procedures Itroducto to Aalyss of Varace Aalyss

More information

2.28 The Wall Street Journal is probably referring to the average number of cubes used per glass measured for some population that they have chosen.

2.28 The Wall Street Journal is probably referring to the average number of cubes used per glass measured for some population that they have chosen. .5 x 54.5 a. x 7. 786 7 b. The raked observatos are: 7.4, 7.5, 7.7, 7.8, 7.9, 8.0, 8.. Sce the sample sze 7 s odd, the meda s the (+)/ 4 th raked observato, or meda 7.8 c. The cosumer would more lkely

More information

Evaluation of uncertainty in measurements

Evaluation of uncertainty in measurements Evaluato of ucertaty measuremets Laboratory of Physcs I Faculty of Physcs Warsaw Uversty of Techology Warszawa, 05 Itroducto The am of the measuremet s to determe the measured value. Thus, the measuremet

More information

1. The weight of six Golden Retrievers is 66, 61, 70, 67, 92 and 66 pounds. The weight of six Labrador Retrievers is 54, 60, 72, 78, 84 and 67.

1. The weight of six Golden Retrievers is 66, 61, 70, 67, 92 and 66 pounds. The weight of six Labrador Retrievers is 54, 60, 72, 78, 84 and 67. Ecoomcs 3 Itroducto to Ecoometrcs Sprg 004 Professor Dobk Name Studet ID Frst Mdterm Exam You must aswer all the questos. The exam s closed book ad closed otes. You may use your calculators but please

More information

Multivariate Transformation of Variables and Maximum Likelihood Estimation

Multivariate Transformation of Variables and Maximum Likelihood Estimation Marquette Uversty Multvarate Trasformato of Varables ad Maxmum Lkelhood Estmato Dael B. Rowe, Ph.D. Assocate Professor Departmet of Mathematcs, Statstcs, ad Computer Scece Copyrght 03 by Marquette Uversty

More information

ρ < 1 be five real numbers. The

ρ < 1 be five real numbers. The Lecture o BST 63: Statstcal Theory I Ku Zhag, /0/006 Revew for the prevous lecture Deftos: covarace, correlato Examples: How to calculate covarace ad correlato Theorems: propertes of correlato ad covarace

More information

STA 108 Applied Linear Models: Regression Analysis Spring Solution for Homework #1

STA 108 Applied Linear Models: Regression Analysis Spring Solution for Homework #1 STA 08 Appled Lear Models: Regresso Aalyss Sprg 0 Soluto for Homework #. Let Y the dollar cost per year, X the umber of vsts per year. The the mathematcal relato betwee X ad Y s: Y 300 + X. Ths s a fuctoal

More information

Linear Regression. Hsiao-Lung Chan Dept Electrical Engineering Chang Gung University, Taiwan

Linear Regression. Hsiao-Lung Chan Dept Electrical Engineering Chang Gung University, Taiwan Lear Regresso Hsao-Lug Cha Dept Electrcal Egeerg Chag Gug Uverst, Tawa chahl@mal.cgu.edu.tw Curve fttg Least-squares regresso Data ehbt a sgfcat degree of error or scatter A curve for the tred of the data

More information

C. Statistics. X = n geometric the n th root of the product of numerical data ln X GM = or ln GM = X 2. X n X 1

C. Statistics. X = n geometric the n th root of the product of numerical data ln X GM = or ln GM = X 2. X n X 1 C. Statstcs a. Descrbe the stages the desg of a clcal tral, takg to accout the: research questos ad hypothess, lterature revew, statstcal advce, choce of study protocol, ethcal ssues, data collecto ad

More information

Bivariate regression. dependent and independent variables. Which way is the relationship?

Bivariate regression. dependent and independent variables. Which way is the relationship? Bvarate regresso the correlato coeffcet measures the assocato betwee sets of pared varates, but t does ot tell us the way the two varables are related does ot allow us to predct the value of oe varable

More information

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Exam: ECON430 Statstcs Date of exam: Frday, December 8, 07 Grades are gve: Jauary 4, 08 Tme for exam: 0900 am 00 oo The problem set covers 5 pages Resources allowed:

More information

{ }{ ( )} (, ) = ( ) ( ) ( ) Chapter 14 Exercises in Sampling Theory. Exercise 1 (Simple random sampling): Solution:

{ }{ ( )} (, ) = ( ) ( ) ( ) Chapter 14 Exercises in Sampling Theory. Exercise 1 (Simple random sampling): Solution: Chapter 4 Exercses Samplg Theory Exercse (Smple radom samplg: Let there be two correlated radom varables X ad A sample of sze s draw from a populato by smple radom samplg wthout replacemet The observed

More information

Chapter 4 Multiple Random Variables

Chapter 4 Multiple Random Variables Revew for the prevous lecture: Theorems ad Examples: How to obta the pmf (pdf) of U = g (, Y) ad V = g (, Y) Chapter 4 Multple Radom Varables Chapter 44 Herarchcal Models ad Mxture Dstrbutos Examples:

More information

Lecture 2: Linear Least Squares Regression

Lecture 2: Linear Least Squares Regression Lecture : Lear Least Squares Regresso Dave Armstrog UW Mlwaukee February 8, 016 Is the Relatoshp Lear? lbrary(car) data(davs) d 150) Davs$weght[d]

More information

CLASS NOTES. for. PBAF 528: Quantitative Methods II SPRING Instructor: Jean Swanson. Daniel J. Evans School of Public Affairs

CLASS NOTES. for. PBAF 528: Quantitative Methods II SPRING Instructor: Jean Swanson. Daniel J. Evans School of Public Affairs CLASS NOTES for PBAF 58: Quattatve Methods II SPRING 005 Istructor: Jea Swaso Dael J. Evas School of Publc Affars Uversty of Washgto Ackowledgemet: The structor wshes to thak Rachel Klet, Assstat Professor,

More information

Chapter Statistics Background of Regression Analysis

Chapter Statistics Background of Regression Analysis Chapter 06.0 Statstcs Backgroud of Regresso Aalyss After readg ths chapter, you should be able to:. revew the statstcs backgroud eeded for learg regresso, ad. kow a bref hstory of regresso. Revew of Statstcal

More information

Lecture Notes Forecasting the process of estimating or predicting unknown situations

Lecture Notes Forecasting the process of estimating or predicting unknown situations Lecture Notes. Ecoomc Forecastg. Forecastg the process of estmatg or predctg ukow stuatos Eample usuall ecoomsts predct future ecoomc varables Forecastg apples to a varet of data () tme seres data predctg

More information

Lecture 8: Linear Regression

Lecture 8: Linear Regression Lecture 8: Lear egresso May 4, GENOME 56, Sprg Goals Develop basc cocepts of lear regresso from a probablstc framework Estmatg parameters ad hypothess testg wth lear models Lear regresso Su I Lee, CSE

More information

Parameter, Statistic and Random Samples

Parameter, Statistic and Random Samples Parameter, Statstc ad Radom Samples A parameter s a umber that descrbes the populato. It s a fxed umber, but practce we do ot kow ts value. A statstc s a fucto of the sample data,.e., t s a quatty whose

More information

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Postpoed exam: ECON430 Statstcs Date of exam: Jauary 0, 0 Tme for exam: 09:00 a.m. :00 oo The problem set covers 5 pages Resources allowed: All wrtte ad prted

More information

Lecture 3. Sampling, sampling distributions, and parameter estimation

Lecture 3. Sampling, sampling distributions, and parameter estimation Lecture 3 Samplg, samplg dstrbutos, ad parameter estmato Samplg Defto Populato s defed as the collecto of all the possble observatos of terest. The collecto of observatos we take from the populato s called

More information

Linear Regression. Hsiao-Lung Chan Dept Electrical Engineering Chang Gung University, Taiwan

Linear Regression. Hsiao-Lung Chan Dept Electrical Engineering Chang Gung University, Taiwan Lear Regresso Hsao-Lug Cha Dept Electrcal Egeerg Chag Gug Uverst, Tawa chahl@mal.cgu.edu.tw Curve fttg Least-squares regresso Data ehbt a sgfcat degree of error or scatter A curve for the tred of the data

More information

Lecture Notes 2. The ability to manipulate matrices is critical in economics.

Lecture Notes 2. The ability to manipulate matrices is critical in economics. Lecture Notes. Revew of Matrces he ablt to mapulate matrces s crtcal ecoomcs.. Matr a rectagular arra of umbers, parameters, or varables placed rows ad colums. Matrces are assocated wth lear equatos. lemets

More information

Chapter 3 Sampling For Proportions and Percentages

Chapter 3 Sampling For Proportions and Percentages Chapter 3 Samplg For Proportos ad Percetages I may stuatos, the characterstc uder study o whch the observatos are collected are qualtatve ature For example, the resposes of customers may marketg surveys

More information

Comparison of Dual to Ratio-Cum-Product Estimators of Population Mean

Comparison of Dual to Ratio-Cum-Product Estimators of Population Mean Research Joural of Mathematcal ad Statstcal Sceces ISS 30 6047 Vol. 1(), 5-1, ovember (013) Res. J. Mathematcal ad Statstcal Sc. Comparso of Dual to Rato-Cum-Product Estmators of Populato Mea Abstract

More information

STA 105-M BASIC STATISTICS (This is a multiple choice paper.)

STA 105-M BASIC STATISTICS (This is a multiple choice paper.) DCDM BUSINESS SCHOOL September Mock Eamatos STA 0-M BASIC STATISTICS (Ths s a multple choce paper.) Tme: hours 0 mutes INSTRUCTIONS TO CANDIDATES Do ot ope ths questo paper utl you have bee told to do

More information

Econometrics. 3) Statistical properties of the OLS estimator

Econometrics. 3) Statistical properties of the OLS estimator 30C0000 Ecoometrcs 3) Statstcal propertes of the OLS estmator Tmo Kuosmae Professor, Ph.D. http://omepre.et/dex.php/tmokuosmae Today s topcs Whch assumptos are eeded for OLS to work? Statstcal propertes

More information

A Combination of Adaptive and Line Intercept Sampling Applicable in Agricultural and Environmental Studies

A Combination of Adaptive and Line Intercept Sampling Applicable in Agricultural and Environmental Studies ISSN 1684-8403 Joural of Statstcs Volume 15, 008, pp. 44-53 Abstract A Combato of Adaptve ad Le Itercept Samplg Applcable Agrcultural ad Evrometal Studes Azmer Kha 1 A adaptve procedure s descrbed for

More information

UNIT 4 SOME OTHER SAMPLING SCHEMES

UNIT 4 SOME OTHER SAMPLING SCHEMES UIT 4 SOE OTHER SAPLIG SCHEES Some Other Samplg Schemes Structure 4. Itroducto Objectves 4. Itroducto to Systematc Samplg 4.3 ethods of Systematc Samplg Lear Systematc Samplg Crcular Systematc Samplg Advatages

More information

Lecture Note to Rice Chapter 8

Lecture Note to Rice Chapter 8 ECON 430 HG revsed Nov 06 Lecture Note to Rce Chapter 8 Radom matrces Let Y, =,,, m, =,,, be radom varables (r.v. s). The matrx Y Y Y Y Y Y Y Y Y Y = m m m s called a radom matrx ( wth a ot m-dmesoal dstrbuto,

More information

Simulation Output Analysis

Simulation Output Analysis Smulato Output Aalyss Summary Examples Parameter Estmato Sample Mea ad Varace Pot ad Iterval Estmato ermatg ad o-ermatg Smulato Mea Square Errors Example: Sgle Server Queueg System x(t) S 4 S 4 S 3 S 5

More information

LINEAR REGRESSION ANALYSIS

LINEAR REGRESSION ANALYSIS LINEAR REGRESSION ANALYSIS MODULE V Lecture - Correctg Model Iadequaces Through Trasformato ad Weghtg Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur Aalytcal methods for

More information