REGRESSION ANALYSIS II- MULTICOLLINEARITY
|
|
- Antony Reed
- 6 years ago
- Views:
Transcription
1 REGRESSION ANALYSIS II- MULTICOLLINEARITY
2 QUESTION 1 Departments of Open Unversty of Cyprus A and B consst of na = 35 and nb = 30 students respectvely. The students of department A acheved an average test score ˆ A 7.5, whle the students of department B acheved an average test score ˆ 6 B (a) If the standard devaton of the department A s known and equals to 2.5, examne the null hypothess that the average test score of the students of A equals to 8.5 versus the alternatve hypothess that t s less than 8.5 (b) If the standard devaton of the department B s unknown whle ts estmate equals to 1.5, examne the null hypothess that the average test score of the students of B equals to 5 versus the alternatve hypothess that t s more than 5
3 () Standard devaton s known Queston 1 We determne the null hypothess Η 0 and the alternatve hypothess Η 1 : Η 0 : μ Α = 8,5 Η 1 : μ Α < 8,5 We choose the sutable test statstc, and we calculate ts value: The sutable test statstc s: We calculate ts value: Z = μ Α 8,5 σ/ n A ~Ν(0,1) Z = μ Α 8,5 σ/ n A = 7,5 8,5 2,5/ 35 = 1 4,23 = 2,366. We determne the acceptance regon C 0 and the rejecton regon C 1 : C 0 Z : Z c and C 1 Z : Z < c.
4 Queston 1 We choose the sgnfcance level α, whch determnes the probablty of commttng Type I error: P Z < c; H 0 s vald = α. Based on the last relatonshp, we fnd from the standard normal dstrbuton table the crtcal value c: Level of sgnfcance Crtcal value 1% -2,33 5% -1,64 10% -1,28 Decson: we have found that Z = 2,366. At level of sgnfcance 1% the crtcal value s c = 2,33. We have Z < c, so we reject Η 0. At level of sgnfcance 5% the crtcal value s c = 1,65. We have Z < c, so we reject Η 0. At level of sgnfcance 10% the crtcal value s c = 1,28. We have Z < c, so we reject Η 0.
5 (b) Queston 1 The standard devaton σ s unknown. We determne the null hypothess Η 0 and the alternatve hypothess Η 1 : Η 0 : μ Β = 5 Η 1 : μ Β > 5 We choose the sutable test statstc, and then we calculate ts value: The sutable test statstc s the t-statstc: t = μ Β 5 s Β / n B ~St(n Β 1). Then we calculate ts value: t = μ Β 5 s Β / n B = 6 5 = 1,0 = 0,122. 1,5 30 8,22 We determne the acceptance regon C 0 and the rejecton regon C 1 : C 0 t : t c και C 1 t : t > c.
6 Queston 1 We select the sgnfcance level α, whch s the probablty to commt Type I error: P t > c; H 0 s vald = α. From the last equaton we fnd the crtcal value c from the Student s t dstrbuton table wth degrees of freedom equal to 30-1 = 29. Snce we have 29 degrees of freedom we get: Level of sgnfcance Crtcal value 1% 2,462 5% 1,699 10% 1,311 Make a Decson: We have found that t = 0,122. At level of sgnfcance 5% the crtcal value s c = 1,699. We have t < c, so we accept Η 0. At level of sgnfcance 1% the crtcal value s c = 2,462. We have t < c, so we accept Η 0. At level of sgnfcance 10% the crtcal value s c = 1,311. We have t < c, so we accept Η 0.
7 Queston 2 An economst evaluates the relatonshp between 5 economc varables. He wants to estmate the multple regresson: yt 1 2xt 3zt 4rt 5mt u t =1,2,, 105. The estmaton output s: yˆ t 0,172 0,264 x (2,604) (0,205) t 0,623z (0,190) 0,195r (0,097) 0,222m (0,131) where standard errors are presented n parentheses. (a) Examne the followng hypotheses: () H0 : 2 0 () H0 : 4 0 H1 : 4 0 () H : 0 H : at least one of,,, (v) H : 0 H : at least one of,, The sgnfcance level s 5%. We are gven the followng two models: (1) yˆ t t t t t t, s R , RSS 51196, 28 22,740, 0,184 z 0,250m R 2 0, 189 s 22, 957 RSS 53232, 45 (0,098) (0,122) (2) yˆ t 0,163 zt 0,210 xt R 2 0, 116 s 25, 834 RSS 52144, 12 (0,075) (0,114) 0 t
8 () We determne the null hypothessη 0 H 0 : 2 0 We choose the sutable test statstc, and then we compute ts value: The sutable test statstc s the t-statstc: t = β 2 β 2 ~St(n k). SE(β 2 ) Ts the number of observatons, so we haven = 105, whleks the number of the parameters n the regresson (k = 5, β 1, β 2, β 3, β 4, β 5 ). Also, SE(β 2 )s the standard error of the regressorβ 2. The value of the t-statstc s computed as: t = 0, ,205 = 1,290. We calculate thep-value as the degree of support ofη 0 : p value = P t t ; H 0 s vald = 2 P t t ; H 0 s vald = = 2 P t 1.290; H 0 s vald = 2 0,10 = 0,20. Thep value = 0,20s very large (larger than 10%) so there s strong support of the null hypothess H 0 : 2 0.Thus,coeffcent 2 s statstcally nsgnfcant (statstcally s equal to zero).
9 () Determne the null hypothess Η 0 and the alternatve hypothess Η 1 : Η 0 : β 4 = 0 Η 1 : β 4 < 0 We choose the sutable test statstc and then we compute ts value: The sutable test statstc s the t-statstc: t = β 4 β ~St(n k), SE(β 4 ) where SE(β 4 ) s the standard error of β 4 and k, s the number of the parameters of the regresson. The test statstc s calculated as t = 0, ,097 = We determne the acceptance regon C 0 and the rejecton regon C 1 : C 0 t : t c and C 1 t : t < c.
10 We select the sgnfcance level α, whch represents the probablty of commttng Type Ι error: P t < c; H 0 s vald = α =>1 P t c; H 0 s vald = α => P t c; H 0 s vald = 1 α Based on the last equaton we get the crtcal value c from the t dstrbuton table t n k = t = t 100. Sgnfcance level Crtcal value 5% Make a decson: We accept Η 0 when the value of the t-statstc, t * «falls» nto the acceptance regon C 0, whle we reject Η 0 and accept Η 1 when t * «falls» ntothe rejecton regon C 1. At sgnfcance level 5% the crtcal valuec = 1,660. But we fnd that t = 2.01 < c. So we reject Η 0 and accept Η 1.
11 () We want to examne whether all parameter coeffcents, except the constant, are smultaneously equal to zero. We determne the null hypothess Η 0 and the alternatve hypothess Η 1 : Η 0 : β 2 = β 3 = β 4 = β 5 = 0 Η 1 : at least one ofβ 2, β 3, β 4, β 5 0 We choose the sutable test statstc and then we compute ts value: The sutable test statstc s the F-statstc F = R2 (n k) ~F(k 1, n k), 1 R 2 (k 1) The value of the F-statstc s calculated as F = 0,219 (105 5) 1 0,219 (5 1) = 0, = 7. We determne the acceptance regon C 0 and the rejecton regon C 1 : C 0 F : F c and C 1 F : F > c.
12 We select the sgnfcance levelα, whch represents the probablty of commttng TypeΙ error: P F > c; H 0 s vald = α From the last equaton we fnd the crtcal value c from the F dstrbuton tablef k 1, n k = F 5 1,105 5 = F(4,100). (We usedf(4,120)becausef(4,100)does not exst n the tables). Level of sgnfcance 5% 2,4472 Crtcal Value Make a decson: We acceptη 0 when the value of the F-statstc, F * «falls» ntothe acceptance regonc 0, whle we rejectη 0 and acceptη 1 whenf * «falls» nto the rejecton regonc 1. For sgnfcance level 5% the crtcal valuec = 2,4472. We fnd thatf = 7 > c. So we rejectη 0 (acceptη 1 ).
13 (v) We want to examne whether parameter coeffcents β1, β2 and β4 are smultaneously equal to zero. If we set smultaneously these coeffcent restrctons, we get the followng restrcted verson of our basc regresson model: y t 3 zt 5 m t u t Thus, between model 1 and 2, we choose model 1 because t corresponds to the restrcted model specfcaton. Estmaton of the restrcted model yelds the followng results: yˆ t 0,184 z (0,098) t 0,250m (0,122) t, R , s , RSS Snce the new coeffcent of determnaton has been calculated for the model under the restrctons we have: R R
14 We determne the null hypothess Η 0 and the alternatve hypothess Η 1 : Η 0 : β 1 = β 2 = β 4 = 0 Η 1 : at least one ofβ 1, β 2, β 4 0 We choose the sutable test statstc, and then we calculate ts value. The sutable test statstc s the F-statstc F = R 2 2 U R R /m 2 ~F(m, n k). 1 R U /(n k) The value of the F-statstc s calculated as: F = 0,219 0, ,219 (105 5) 3 = 1,28. We determne the acceptance regon C 0 and the rejecton regon C 1 : C 0 F : F c and C 1 F : F > c.
15 We choose the level of sgnfcance α, whch represents the probablty of commttng TypeΙ error: P F > c; H 0 s vald = α Based on the last equaton we fnd the crtcal value c from the F dstrbuton table F m, n k = F(3,105 5). Level of Sgnfcance 5% 2,6802 Crtcal Value Make a decson: We accept Η 0 when the value of the F-statstc, F * «falls» nto the acceptance regon C 0, whle we reject Η 0 and accept the alternatve hypothess Η 1 when F * «falls» nto the rejecton regon C 1. For sgnfcance level 5%, the crtcal value s c = 2,6802. We fnd that F = 1,28 < c. So we accept Η 0 (and reject Η 1 ).
16 Queston 3 A random sample of 1,562 persons was asked to respond on a scale from one (strongly dsagree) to seven (strongly agree) to the queston: Wll the new government economc polcy lower unemployment?. The sample mean response was 4.27 and the populaton standard devaton was Test whether the mean response s equal to 4 aganst the alternatve that t s dfferent than 4. Perform the hypothess test at level 5%.
17 We determne the null hypothess Η 0 and the alternatve hypothess Η 1 : Η 0 : μ = 4 Η 1 : μ 4 We choose the sutable test statstc, and we calculate ts value. The sutable test statstc s the z-statstc because we have a large sample sze: Z = μ 4 σ/ n A ~Ν(0,1) We calculate ts value: Z = μ 4 = σ/ n A 1.32/ 1562 = We determne the acceptance regon C 0 and the rejecton regon C 1 : C 0 Z : Z < c and C 1 Z : Z c.
18 We choose the sgnfcance level α, whch determnes the probablty of commttng Type I error: * P( Z c; H0 s vald ) 2* P( Z P( Z * * c; H c; H 0 0 s vald ) s vald ) / 2 c z / 2 Based on the last relatonshp, we fnd from the standard normal dstrbuton table the crtcal value c: Level of sgnfcance Crtcal value 5% 1.96 Make a decson: we have found that Z = At level of sgnfcance 5%, the crtcal value s c = We have Z > c, so we reject Η 0.
19 Queston 4 Much research n appled economcs focuses on the prcng of goods/servces. One common approach nvolves buldng a model n whch the prce of a good depends on specfc characterstcs of that good. A real estate agent n Canada s nterested n buldng a prcng model for house prces. An approach s to estmate a multple regresson model, where the sales prce of the house n Canadan dollars s the dependent varable Y, whle varous determnants of house prces are used as ndependent varables. Factors whch affect the house prces are the followng: X1 = the lot sze of the property (n square feet) X2 = the number of bedrooms X3 = the number of bathrooms X4 = the number of storeys (excludng the basement). X5 = basement (f the house has a basement) X6 = ar condtonng system (f the house ncludes an ar condtoner) X7 = garage (number of rooms used for storage of vehcles)
20 Data on the housng market of Wndsor of Canada sale prce lot sze bedroom bath storeys basement ar cond garage Data taken from Gary Koop s book Analyss of economc data
21 Queston 4 Ft the regresson model: y x x x x x x x u , 1,2,...,39 Wrte the ftted regresson equaton. If we consder comparable houses, how much would an extra bathroom add to the value of the house? Examne whether the coeffcent β3 s statstcally sgnfcant at level 5%. Test whether all the determnants of the house prces are smultaneously equal to zero aganst the alternatve hypothess that at least one of them s dfferent from zero (at level 5%) Test whether the varables X2, X6 and X7 are smultaneously equal to zero aganst the alternatve hypothess that at least one of them s dfferent from zero (at level 5%)
22 Queston 4 Wrte the ftted regresson equaton We estmate the regresson model by usng the excel functon Regresson. The Estmaton Output s gven below: SUMMARY OUTPUT Regresson Statstcs Multple R R Square Adjusted R Square Standard Error Observatons 39 ANOVA df SS MS F Sgnfcance F Regresson E Resdual E+08 Total Coeffcents Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept X Varable X Varable X Varable X Varable X Varable X Varable X Varable The ftted regresson equaton s: ˆ y x x x x 3241x x x 7
23 Queston 4 If we consder comparable houses, how much would an extra bathroom add to the value of the house? Houses wth an extra bathroom wll worth bˆ Canadan dollars more than those wthout an extra bathroom, f we consder houses wth the same lot sze, number of bedrooms, storeys, basement, etc. The coeffcent estmate of varable X3 measures how much Y wll change when X3 changes one unt, gven that all the other explanatory varables reman the same. ˆb 3 In the case of smple regresson we can say that β measures the nfluence of X on Y ; n the multple regresson we say that βj measures the nfluence of Xj on Y all other explanatory varables beng equal.
24 Economc nterpretaton of the regresson estmates Some ways of verbally statng what the value of β1 means: An extra square foot of lot sze wll tend to add another $6.15 on to the prce of a house, ceters parbus. If we consder houses wth the same number of bedrooms, bathrooms, storeys, etc, an extra square foot of lot sze wll tend to add another $6.15 onto the prce of the house. If we compare houses wth the same number of bedrooms, bathrooms, storeys, etc, those wth larger lots tend to be worth more. In partcular, an extra square foot of lot sze s assocated wth an ncreased prce of $6.15. We cannot smply say that houses wth bgger lots are worth more snce ths s not the case (e.g. some nce houses on small lots wll be worth more than poor houses on large lots). However, we can say that f we consder houses that vary n lot sze, but are comparable n other respects, those wth larger lots tend to be worth more.
25 Examne whether the coeffcent β3 s statstcally sgnfcant at level 5%. The null hypothess Η 0 s H : We choose the sutable test statstc, and then we compute ts value: The sutable test statstc s the t-statstc: t = β 3 β 3 ~St(n k). SE(β 3 ) T s the number of observatons, so we have n = 39, whle k s the number of the parameters n the regresson (k = 8, β 0, β 1, β 2, β 3, β 4, β 5, β 6, β 7 ). Also, SE(β 3 )s the standard error of the regressor β 3. The value of the t-statstc s computed as: t = 14787, ,28 = 2,169. We calculate thep-value as the degree of support ofη 0 : p value = P t t ; H 0 s vald = 2 P t t ; H 0 s vald = = 2 P t 2,169; H 0 s vald = The p value = 0,038 s very small (smaller than 5%) so there s no support of the null hypothess H : Thus, the coeffcent s statstcally sgnfcant (statstcally s not 3 equal to zero).
26 Test whether all the determnants of the house prces are smultaneously equal to zero aganst the alternatve hypothess that at least one of them s dfferent from zero (at level 5%) We determne the null hypothess Η 0 and the alternatve hypothess Η 1 : Η 0 : β 1 = β 2 = β 3 = β 4 = β 5 = β 6 = β 7 = 0 Η 1 : at least one ofβ 1, β 2,.., β 7 0 We choose the sutable test statstc and then we compute ts value: The sutable test statstc s the F-statstc F = R2 (n k) ~F(k 1, n k), 1 R 2 (k 1) The value of the F-statstc s calculated as F = 0,59 (39 8) 1 0,59 (8 1) = = 6.38 We determne the acceptance regon C 0 and the rejecton regon C 1 : C 0 F : F c and C 1 F : F > c.
27 We select the sgnfcance levelα, whch represents the probablty of commttng TypeΙ error: P F > c; H 0 s vald = α From the last equaton we fnd the crtcal value c from the F dstrbuton tablef k 1, n k = F 8 1,39 8 = F(7,31). (We used F(7,30)because F 7,31 does not exst n the tables). Level of sgnfcance 5% Crtcal Value Make a decson: We accept Η 0 when the value of the F-statstc, F * «falls» nto the acceptance regon C 0, whle we reject Η 0 and accept Η 1 when F * «falls» nto the rejecton regon C 1. For sgnfcance level 5% the crtcal valuec = 2,3343. We fnd that F = 6.38 > c. So we reject Η 0 (accept Η 1 ).
28 Test whether the varables X2, X6 and X7 are smultaneously equal to zero aganst the alternatve hypothess that at least one of them s dfferent from zero (at level 5%) If we set smultaneously these coeffcent restrctons, we get the followng restrcted verson of the regresson model: y x x x x u, ,2,...,39 We estmate the new regresson model, and we get the followng results: SUMMARY OUTPUT Regresson Statstcs Multple R R Square Adjusted R Square Standard Error Observatons 39 ANOVA df SS MS F Sgnfcance F Regresson E E-06 Resdual E+08 Total Coeffcents Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept X Varable X Varable X Varable X Varable The new coeffcent of determnaton s the restrcted coeffcent of determnaton: R R
29 We determne the null hypothess Η 0 and the alternatve hypothess Η 1 : Η 0 : β 2 = β 6 = β 7 = 0 Η 1 : at least one ofβ 2, β 6, β 7 0 We choose the sutable test statstc, and then we calculate ts value. The sutable test statstc s the F-statstc F = R 2 2 U R R /m 2 ~F(m, n k). 1 R U /(n k) The value of the F-statstc s calculated as: F = 0,5904 0, ,5904 (39 8) 3 = We determne the acceptance regon C 0 and the rejecton regon C 1 : C 0 F : F c and C 1 F : F > c.
30 We choose the level of sgnfcance α, whch represents the probablty of commttng TypeΙ error: P F > c; H 0 s vald = α Based on the last equaton we fnd the crtcal value c from the F dstrbuton tablef m, n k = F(3,39 8). (We used F(3,30) because F 3,31 does not exst n the tables). Level of Sgnfcance 5% 2,9223 Crtcal Value Make a decson: We accept Η 0 when the value of the F-statstc, F * «falls» nto the acceptance regon C 0, whle we reject Η 0 and accept the alternatve hypothess Η 1 when F * «falls» nto the rejecton regon C 1. For sgnfcance level 5%, the crtcal value s c = 2,9223. We fnd that F = < c. So we accept Η 0 (and reject Η 1 ).
31 Ptfalls of usng multple regresson analyss In multple regresson analyss, we are usually facng two types of problems: The Effect of Includng a Varable that Ought not to be Included The Omtted varables bas The Effect of Includng a Varable that Ought not to be Included If we nclude explanatory varables that should not be present n the regresson, then the estmated coeffcents on the varables wll not be accurate In the prevous example, addng an extra bedroom to the house wll rase ts prce by $14,787.69? Probably Not! The reason s that there are many factors other than the number of bedrooms that potentally nfluence house prces. (for example, bathrooms or lot sze are more mportant determnants of house prces than bedrooms. ) Furthermore, these factors may be hghly correlated (.e. houses wth more bathrooms tend to have more bedrooms). To nvestgate the possblty, let us examne the correlaton matrx of all the varables n the model
32 Correlaton Matrx of the varables We calculate the correlaton coeffcent between each par of varables, and then we present the results n a matrx. For example, f we have three varables, X, Y and Z, then there are three possble correlatons (.e. ρxy, ρxz and ρyz ). Then we put these correlatons n a matrx: X Y Z X 1 Y ρxy 1 Z ρxz ρyz 1 You can use the excel functon Correlaton n the Data Analyss Toolbox to compute the correlaton matrx of the varables.
33 Correlaton Matrx of the varables In the house prcng regresson model, the correlaton matrx of all varables s the followng: Y X1 X2 X3 X4 X5 X6 X7 Y 1 X X X X X X X Snce all the elements of the correlaton matrx are postve, t follows that each par of varables s postvely correlated wth each other. The correlaton between the number of bathrooms and the number of bedrooms s 0.335, ndcatng that houses wth more bathrooms also tend to have more bedrooms Also note that the correlaton between the number of storeys and the number of bedrooms s 0.536, ndcatng that houses wth more storeys also tend to have more bedrooms. Snce we have found that these factors are hghly correlated wth the bedroom factor, whle t s found to be nsgnfcant, we must exclude t from the regresson model.
34 Multcollnearty When the explanatory varables are very hghly correlated wth each other (correlaton coeffcents ether very close to 1 or to -1) then the problem of multcollnearty occurs. Perfect multcollnearty = under Perfect Multcollnearty, the OLS estmators smply do not exst Imperfect multcollnearty Imperfect multcollnearty (or near multcollnearty) exsts when the explanatory varables n an equaton are correlated, but ths correlaton s less than perfect. In cases of mperfect multcollnearty, the OLS estmators can be obtaned However, the OLS varances are often larger than those obtaned n the absence of multcollnearty.
35 Detectng multcollnearty Auxlary regressons We can determne the relatonshp between any of the regressors and the other regressors by examnng each of the regressors as dependent varables, determne the R 2 values for these regressons, and usng a test to determne the relatonshp between each regressor and the set of other explanatory varables. For example we can run the followng auxlary regresson (for the varable x2): x2 0 1x1 2x3 3x4 4x5 5x6 6x7 u, 1,2,...,39 We wll set up an F test to determne f there s a hgh level of multcollnearty. So we wll test the null hypothess H 0 : aganst the alternatve hypothess that at least one of these coeffcents s dfferent than zero. 0
36 Detectng multcollnearty Auxlary regressons The test statstc wll be a F statstc, whch follows an F dstrbuton wth k-2 and n-k+1 degrees of freedom, where k=the number of explanatory varables, ncludng the ntercept. If F s sgnfcant, t s taken to mean that the partcular X s collnear wth other X's; f the F value s not sgnfcant, the X (as dependent varable) s not consdered to be collnear wth the other explanatory varables. If F s sgnfcant, you may wsh to exclude that varable from the model, snce the part of the dependent varable that t s explanng s already beng explaned by the other explanatory varables. You wll have to determne f t s wse to use a more parsmonous model or not.
37 Detectng multcollnearty Auxlary regressons Back to our example. We run an auxlary regresson for the varable x1. Thus, we use x1 as a dependent varable, and the other regressors as ndependent varables. The estmaton output of the auxlary regresson s presented below:
38 Detectng multcollnearty Auxlary regressons The F statstc s not sgnfcant (ts p-value s , much larger than 0.05), therefore the varable x1 s not consdered to be collnear wth the other explanatory varables.
39 Auxlary regressons Detectng multcollnearty
40 Auxlary regressons Detectng multcollnearty We estmated sx addtonal auxlary regressons wth the remanng varables as dependent varables. The F statstc s found to be sgnfcant for the varables bedroom, bathroom, storeys, and garage, therefore these varables are collnear wth the other explanatory varables. We may wsh to exclude these varables from the model.
41 Detectng multcollnearty Klen's Rule of Thumb suggests that multcollnearty may be a problem only f the R 2 obtaned from an auxlary regresson s greater than the overall R 2 (on the regresson wth y as the dependent varable). In ths example, the overall R 2 s equal to 0.59 whle the R 2 obtaned from the auxlary regressons range from 0.18 to Therefore, accordng to Klen s rule of thumb, there s no problem of multcollnearty snce the auxlary R 2 are smaller than the overall R 2
42 Detectng multcollnearty Egenvalues and Condton Index We can get the egenvalues and the condton ndex to estmate the level of collnearty n the explanatory varables. Most multvarate statstcal approaches nvolve decomposng a correlaton matrx nto lnear combnatons of varables. The lnear combnatons are chosen so that the frst combnaton has the largest possble varance (subject to some restrctons we won't dscuss), the second combnaton has the next largest varance, subject to beng uncorrelated wth the frst, the thrd has the largest possble varance, subject to beng uncorrelated wth the frst and second, and so forth. The varance of each of these lnear combnatons s called an egenvalue.
43 Detectng multcollnearty Egenvalues and Condton Index Number stands for lnear combnaton of X varables. Egenval(ue) stands for the varance of that combnaton. The condton ndex s a smple functon of the egenvalues, namely, CI max where λ s the symbol for an egenvalue.
44 Detectng multcollnearty Egenvalues and Condton Index The fourth part of the matrx s the Varance proportons. Ths s the regresson coeffcent varance- decomposton matrx, whch shows the proporton of varance for each regresson coeffcent (and ts assocated varable) attrbutable to each condton ndex.
45 Detectng multcollnearty Egenvalues and Condton Index To use the table, you frst look at the varance proportons. For X1, for example, most of the varance (about 75 percent) s assocated wth Number 3, whch has an egenvalue of.079 and a condton ndex of Most of the rest of X1 s assocated wth Number 4. Varable X2 s assocated wth 3 dfferent numbers (2, 3, & 4), and X3 s mostly assocated wth Number 2. Look for varance proportons about.50 and larger. Collnearty s spotted by fndng 2 or more varables that have large proportons of varance (.50 or more) that correspond to large condton ndces (between 10 and 30). There s no evdent problem wth collnearty n the above example
46 Detectng multcollnearty Egenvalues and Condton Index Gretl: Clck on Analyss on the estmated model and then select Collnearty : Frst, a threshold of 10 for the condton ndex selects three condton ndexes (10.446, and ) Condton ndex equal to s assocated only to one large varance proporton ( the lotsze varable); thus no collnearty s shown for ths ndex. Condton ndex equal to s assocated wth two large varance proportons (0.38 and bedroom and bath respectvely); thus, there s collnearty between these two varables The last condton ndex (18.667) s relatvely assocated wth the varables lotsze and bedroom (varance proportons and 0.436) ; no collnearty exsts.
47 Detectng multcollnearty Varance Inflaton Factors (VIF) A varance nflaton factor (VIF) quantfes how much the varance of the estmated coeffcent s nflated. The standard errors and hence the varances of the estmated coeffcents are nflated (.e. ncreased) when multcollnearty exsts. So, the varance nflaton factor for the estmated coeffcent b k denoted VIF k s just the factor by whch the varance s nflated. The VIF for the estmated coeffcent b k s calculated as 2 R k VIF k 1 1 R where s the R 2 -value obtaned by regressng the k th predctor on the remanng predctors. A VIF of 1 means that there s no correlaton among the k th predctor and the remanng predctor varables, and hence the varance of b k s not nflated at all. The general rule of thumb s that VIFs exceedng 4 warrant further nvestgaton, whle VIFs exceedng 10 are sgns of serous multcollnearty requrng correcton. 2 k
48 Detectng multcollnearty Gretl: Clck on Analyss on the estmated model and then select Collnearty : VIFs In ths example, there are no sgns of serous multcollnearty because all VIFs are smaller than 10.
49 Resolvng multcollnearty The easest ways to cure the problems are remove one of the collnear varables transform the hghly correlated varables nto a rato go out and collect more data swtch to a hgher frequency In order to reduce the multcollnearty that exsts, t s not suffcent to go out and just collect older data observatons. The data have to be collected n such a way to ensure that the correlatons among the volatng predctors s actually reduced. That s, collectng more of the same knd of data won't help to reduce the multcollnearty.
50 Resolvng multcollnearty Orthogonal auxlary varables Suppose we have three auxlary varables, X1, X2, and X3, whle we have found that X1 s collnear wth X2 and X3. One way to resolve the problem s to drop X1 from the regresson. If we wsh to keep X1 n the model, together wth X2 and X3, we have to transform X1 n a way that s no longer collnear wth these varables. One way s to make X1 orthogonal to X2 and X3. How do we do t? Frst, we can run the followng auxlary regresson by least squares: X 0 1X 2 2X3 1 u Then we keep the resduals u from the prevous model, and we make the followng transformaton: ~ X 1 0 u ~ ~ where X denotes the orthogonal X1. We are now able to use n our basc 1 X1 model as a regressor.
51 Resolvng multcollnearty Orthogonal auxlary varables prevous example: we found that the varable garage s collnear wth the other varables. If we want to keep the varable n our model, we can make t orthogonal to the other factors. We run the regresson where garage s the dependent varable, whle the remanng predctors are used as ndependent varables: Note that the constant of the model s equal to
52 Resolvng multcollnearty Orthogonal auxlary varables Based on ths model, we can generate the resduals. Remember that X X X X X X Y Y u ˆ... ˆ ˆ ˆ ˆ ˆ X X X X ˆ... ˆ ˆ ˆ ˆ X X u 7 7 ˆ We can calculate the resduals of the model n excel:
53 Resolvng multcollnearty Orthogonal auxlary varables Based on ths model, we can generate the resduals. Remember that X X X X X X Y Y u ˆ... ˆ ˆ ˆ ˆ ˆ Alternatvely, we can calculate the resduals of the model n Gretl. Select Save, and then Resduals. The seres of the resduals wll appear as a new varable (n ths case they appear as uhat1).
54 Resolvng multcollnearty Orthogonal auxlary varables The last step s to calculate the orthogonal varable, by usng the formula: X 7 0 u Therefore, we sum each resdual wth the estmated ntercept of the model. Excel: use a new column where you wll add the number to the column of the resduals Gretl: select Add, then defne new varable, and type nsde the box garage_orth = uhat1. The new varable wll appear n the worksheet.
55 The Omtted varables bas If we omt explanatory varables that should be present n the regresson, then the estmated coeffcents on the ncluded varables wll be not accurate. The ntuton behnd why the omsson of varables causes bas s provded n the prevous example: lot sze s an mportant factor for house prces, and thus wants to enter nto the regresson. If we omt t from the regresson, t wll try to enter n the only way t can through ts postve correlaton wth the explanatory varable: number of bedrooms. One practcal consequence of omtted varables bas s that you should always try to nclude all those explanatory varables that could affect the dependent varable. Unfortunately, n practce, ths s rarely possble. House prces, for nstance, depend on many other explanatory varables than those found n the data set (e.g. the state of repar of the house, how pleasant the neghbors are, closet and storage space, whether the house has hardwood floors, the qualty of the garden, etc.). many of the omtted factors wll be subjectve (e.g. how do you measure pleasantness of the neghbors?).
Statistics for Economics & Business
Statstcs for Economcs & Busness Smple Lnear Regresson Learnng Objectves In ths chapter, you learn: How to use regresson analyss to predct the value of a dependent varable based on an ndependent varable
More informationChapter 13: Multiple Regression
Chapter 13: Multple Regresson 13.1 Developng the multple-regresson Model The general model can be descrbed as: It smplfes for two ndependent varables: The sample ft parameter b 0, b 1, and b are used to
More informationDepartment of Quantitative Methods & Information Systems. Time Series and Their Components QMIS 320. Chapter 6
Department of Quanttatve Methods & Informaton Systems Tme Seres and Ther Components QMIS 30 Chapter 6 Fall 00 Dr. Mohammad Zanal These sldes were modfed from ther orgnal source for educatonal purpose only.
More informationStatistics for Business and Economics
Statstcs for Busness and Economcs Chapter 11 Smple Regresson Copyrght 010 Pearson Educaton, Inc. Publshng as Prentce Hall Ch. 11-1 11.1 Overvew of Lnear Models n An equaton can be ft to show the best lnear
More informationChapter 11: Simple Linear Regression and Correlation
Chapter 11: Smple Lnear Regresson and Correlaton 11-1 Emprcal Models 11-2 Smple Lnear Regresson 11-3 Propertes of the Least Squares Estmators 11-4 Hypothess Test n Smple Lnear Regresson 11-4.1 Use of t-tests
More information[ ] λ λ λ. Multicollinearity. multicollinearity Ragnar Frisch (1934) perfect exact. collinearity. multicollinearity. exact
Multcollnearty multcollnearty Ragnar Frsch (934 perfect exact collnearty multcollnearty K exact λ λ λ K K x+ x+ + x 0 0.. λ, λ, λk 0 0.. x perfect ntercorrelated λ λ λ x+ x+ + KxK + v 0 0.. v 3 y β + β
More informationStatistics for Managers Using Microsoft Excel/SPSS Chapter 13 The Simple Linear Regression Model and Correlation
Statstcs for Managers Usng Mcrosoft Excel/SPSS Chapter 13 The Smple Lnear Regresson Model and Correlaton 1999 Prentce-Hall, Inc. Chap. 13-1 Chapter Topcs Types of Regresson Models Determnng the Smple Lnear
More informationx i1 =1 for all i (the constant ).
Chapter 5 The Multple Regresson Model Consder an economc model where the dependent varable s a functon of K explanatory varables. The economc model has the form: y = f ( x,x,..., ) xk Approxmate ths by
More informationComparison of Regression Lines
STATGRAPHICS Rev. 9/13/2013 Comparson of Regresson Lnes Summary... 1 Data Input... 3 Analyss Summary... 4 Plot of Ftted Model... 6 Condtonal Sums of Squares... 6 Analyss Optons... 7 Forecasts... 8 Confdence
More informationJanuary Examinations 2015
24/5 Canddates Only January Examnatons 25 DO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR STUDENT CANDIDATE NO.. Department Module Code Module Ttle Exam Duraton (n words)
More informationPsychology 282 Lecture #24 Outline Regression Diagnostics: Outliers
Psychology 282 Lecture #24 Outlne Regresson Dagnostcs: Outlers In an earler lecture we studed the statstcal assumptons underlyng the regresson model, ncludng the followng ponts: Formal statement of assumptons.
More informationChapter 15 - Multiple Regression
Chapter - Multple Regresson Chapter - Multple Regresson Multple Regresson Model The equaton that descrbes how the dependent varable y s related to the ndependent varables x, x,... x p and an error term
More informationBasic Business Statistics, 10/e
Chapter 13 13-1 Basc Busness Statstcs 11 th Edton Chapter 13 Smple Lnear Regresson Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc. Chap 13-1 Learnng Objectves In ths chapter, you learn: How to use regresson
More informationChapter 15 Student Lecture Notes 15-1
Chapter 15 Student Lecture Notes 15-1 Basc Busness Statstcs (9 th Edton) Chapter 15 Multple Regresson Model Buldng 004 Prentce-Hall, Inc. Chap 15-1 Chapter Topcs The Quadratc Regresson Model Usng Transformatons
More informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 30 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 2 Remedes for multcollnearty Varous technques have
More informationDO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR. Introductory Econometrics 1 hour 30 minutes
25/6 Canddates Only January Examnatons 26 Student Number: Desk Number:...... DO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR Department Module Code Module Ttle Exam Duraton
More informationScatter Plot x
Construct a scatter plot usng excel for the gven data. Determne whether there s a postve lnear correlaton, negatve lnear correlaton, or no lnear correlaton. Complete the table and fnd the correlaton coeffcent
More information[The following data appear in Wooldridge Q2.3.] The table below contains the ACT score and college GPA for eight college students.
PPOL 59-3 Problem Set Exercses n Smple Regresson Due n class /8/7 In ths problem set, you are asked to compute varous statstcs by hand to gve you a better sense of the mechancs of the Pearson correlaton
More information2016 Wiley. Study Session 2: Ethical and Professional Standards Application
6 Wley Study Sesson : Ethcal and Professonal Standards Applcaton LESSON : CORRECTION ANALYSIS Readng 9: Correlaton and Regresson LOS 9a: Calculate and nterpret a sample covarance and a sample correlaton
More informationECONOMICS 351*-A Mid-Term Exam -- Fall Term 2000 Page 1 of 13 pages. QUEEN'S UNIVERSITY AT KINGSTON Department of Economics
ECOOMICS 35*-A Md-Term Exam -- Fall Term 000 Page of 3 pages QUEE'S UIVERSITY AT KIGSTO Department of Economcs ECOOMICS 35* - Secton A Introductory Econometrcs Fall Term 000 MID-TERM EAM ASWERS MG Abbott
More informationSTAT 3008 Applied Regression Analysis
STAT 3008 Appled Regresson Analyss Tutoral : Smple Lnear Regresson LAI Chun He Department of Statstcs, The Chnese Unversty of Hong Kong 1 Model Assumpton To quantfy the relatonshp between two factors,
More informationThe Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
More informationTests of Single Linear Coefficient Restrictions: t-tests and F-tests. 1. Basic Rules. 2. Testing Single Linear Coefficient Restrictions
ECONOMICS 35* -- NOTE ECON 35* -- NOTE Tests of Sngle Lnear Coeffcent Restrctons: t-tests and -tests Basc Rules Tests of a sngle lnear coeffcent restrcton can be performed usng ether a two-taled t-test
More informationStatistics II Final Exam 26/6/18
Statstcs II Fnal Exam 26/6/18 Academc Year 2017/18 Solutons Exam duraton: 2 h 30 mn 1. (3 ponts) A town hall s conductng a study to determne the amount of leftover food produced by the restaurants n the
More informationCorrelation and Regression. Correlation 9.1. Correlation. Chapter 9
Chapter 9 Correlaton and Regresson 9. Correlaton Correlaton A correlaton s a relatonshp between two varables. The data can be represented b the ordered pars (, ) where s the ndependent (or eplanator) varable,
More informationEcon107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)
I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes
More informationResource Allocation and Decision Analysis (ECON 8010) Spring 2014 Foundations of Regression Analysis
Resource Allocaton and Decson Analss (ECON 800) Sprng 04 Foundatons of Regresson Analss Readng: Regresson Analss (ECON 800 Coursepak, Page 3) Defntons and Concepts: Regresson Analss statstcal technques
More informationNegative Binomial Regression
STATGRAPHICS Rev. 9/16/2013 Negatve Bnomal Regresson Summary... 1 Data Input... 3 Statstcal Model... 3 Analyss Summary... 4 Analyss Optons... 7 Plot of Ftted Model... 8 Observed Versus Predcted... 10 Predctons...
More informationEconomics 130. Lecture 4 Simple Linear Regression Continued
Economcs 130 Lecture 4 Contnued Readngs for Week 4 Text, Chapter and 3. We contnue wth addressng our second ssue + add n how we evaluate these relatonshps: Where do we get data to do ths analyss? How do
More informationLecture Notes for STATISTICAL METHODS FOR BUSINESS II BMGT 212. Chapters 14, 15 & 16. Professor Ahmadi, Ph.D. Department of Management
Lecture Notes for STATISTICAL METHODS FOR BUSINESS II BMGT 1 Chapters 14, 15 & 16 Professor Ahmad, Ph.D. Department of Management Revsed August 005 Chapter 14 Formulas Smple Lnear Regresson Model: y =
More informationY = β 0 + β 1 X 1 + β 2 X β k X k + ε
Chapter 3 Secton 3.1 Model Assumptons: Multple Regresson Model Predcton Equaton Std. Devaton of Error Correlaton Matrx Smple Lnear Regresson: 1.) Lnearty.) Constant Varance 3.) Independent Errors 4.) Normalty
More information1. Inference on Regression Parameters a. Finding Mean, s.d and covariance amongst estimates. 2. Confidence Intervals and Working Hotelling Bands
Content. Inference on Regresson Parameters a. Fndng Mean, s.d and covarance amongst estmates.. Confdence Intervals and Workng Hotellng Bands 3. Cochran s Theorem 4. General Lnear Testng 5. Measures of
More informationModule Contact: Dr Susan Long, ECO Copyright of the University of East Anglia Version 1
UNIVERSITY OF EAST ANGLIA School of Economcs Man Seres PG Examnaton 016-17 ECONOMETRIC METHODS ECO-7000A Tme allowed: hours Answer ALL FOUR Questons. Queston 1 carres a weght of 5%; Queston carres 0%;
More informationChapter 14 Simple Linear Regression
Chapter 4 Smple Lnear Regresson Chapter 4 - Smple Lnear Regresson Manageral decsons often are based on the relatonshp between two or more varables. Regresson analss can be used to develop an equaton showng
More informationx = , so that calculated
Stat 4, secton Sngle Factor ANOVA notes by Tm Plachowsk n chapter 8 we conducted hypothess tests n whch we compared a sngle sample s mean or proporton to some hypotheszed value Chapter 9 expanded ths to
More informationDepartment of Statistics University of Toronto STA305H1S / 1004 HS Design and Analysis of Experiments Term Test - Winter Solution
Department of Statstcs Unversty of Toronto STA35HS / HS Desgn and Analyss of Experments Term Test - Wnter - Soluton February, Last Name: Frst Name: Student Number: Instructons: Tme: hours. Ads: a non-programmable
More informationSTATISTICS QUESTIONS. Step by Step Solutions.
STATISTICS QUESTIONS Step by Step Solutons www.mathcracker.com 9//016 Problem 1: A researcher s nterested n the effects of famly sze on delnquency for a group of offenders and examnes famles wth one to
More informationa. (All your answers should be in the letter!
Econ 301 Blkent Unversty Taskn Econometrcs Department of Economcs Md Term Exam I November 8, 015 Name For each hypothess testng n the exam complete the followng steps: Indcate the test statstc, ts crtcal
More informationIntroduction to Regression
Introducton to Regresson Dr Tom Ilvento Department of Food and Resource Economcs Overvew The last part of the course wll focus on Regresson Analyss Ths s one of the more powerful statstcal technques Provdes
More informationChapter 3. Two-Variable Regression Model: The Problem of Estimation
Chapter 3. Two-Varable Regresson Model: The Problem of Estmaton Ordnary Least Squares Method (OLS) Recall that, PRF: Y = β 1 + β X + u Thus, snce PRF s not drectly observable, t s estmated by SRF; that
More informationLecture 6: Introduction to Linear Regression
Lecture 6: Introducton to Lnear Regresson An Manchakul amancha@jhsph.edu 24 Aprl 27 Lnear regresson: man dea Lnear regresson can be used to study an outcome as a lnear functon of a predctor Example: 6
More informationOutline. Zero Conditional mean. I. Motivation. 3. Multiple Regression Analysis: Estimation. Read Wooldridge (2013), Chapter 3.
Outlne 3. Multple Regresson Analyss: Estmaton I. Motvaton II. Mechancs and Interpretaton of OLS Read Wooldrdge (013), Chapter 3. III. Expected Values of the OLS IV. Varances of the OLS V. The Gauss Markov
More informationStatistics for Managers Using Microsoft Excel/SPSS Chapter 14 Multiple Regression Models
Statstcs for Managers Usng Mcrosoft Excel/SPSS Chapter 14 Multple Regresson Models 1999 Prentce-Hall, Inc. Chap. 14-1 Chapter Topcs The Multple Regresson Model Contrbuton of Indvdual Independent Varables
More informationCorrelation and Regression
Correlaton and Regresson otes prepared by Pamela Peterson Drake Index Basc terms and concepts... Smple regresson...5 Multple Regresson...3 Regresson termnology...0 Regresson formulas... Basc terms and
More informationLinear Regression Analysis: Terminology and Notation
ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page ) Lnear Regresson Analyss: Termnology and Notaton Consder the generc verson of the smple (two-varable) lnear regresson model. It s represented
More informationANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)
Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of
More informationECONOMETRICS - FINAL EXAM, 3rd YEAR (GECO & GADE)
ECONOMETRICS - FINAL EXAM, 3rd YEAR (GECO & GADE) June 7, 016 15:30 Frst famly name: Name: DNI/ID: Moble: Second famly Name: GECO/GADE: Instructor: E-mal: Queston 1 A B C Blank Queston A B C Blank Queston
More information18. SIMPLE LINEAR REGRESSION III
8. SIMPLE LINEAR REGRESSION III US Domestc Beers: Calores vs. % Alcohol Ftted Values and Resduals To each observed x, there corresponds a y-value on the ftted lne, y ˆ ˆ = α + x. The are called ftted values.
More informationPolynomial Regression Models
LINEAR REGRESSION ANALYSIS MODULE XII Lecture - 6 Polynomal Regresson Models Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur Test of sgnfcance To test the sgnfcance
More informationHere is the rationale: If X and y have a strong positive relationship to one another, then ( x x) will tend to be positive when ( y y)
Secton 1.5 Correlaton In the prevous sectons, we looked at regresson and the value r was a measurement of how much of the varaton n y can be attrbuted to the lnear relatonshp between y and x. In ths secton,
More informationThe Ordinary Least Squares (OLS) Estimator
The Ordnary Least Squares (OLS) Estmator 1 Regresson Analyss Regresson Analyss: a statstcal technque for nvestgatng and modelng the relatonshp between varables. Applcatons: Engneerng, the physcal and chemcal
More informationNANYANG TECHNOLOGICAL UNIVERSITY SEMESTER I EXAMINATION MTH352/MH3510 Regression Analysis
NANYANG TECHNOLOGICAL UNIVERSITY SEMESTER I EXAMINATION 014-015 MTH35/MH3510 Regresson Analyss December 014 TIME ALLOWED: HOURS INSTRUCTIONS TO CANDIDATES 1. Ths examnaton paper contans FOUR (4) questons
More informationFirst Year Examination Department of Statistics, University of Florida
Frst Year Examnaton Department of Statstcs, Unversty of Florda May 7, 010, 8:00 am - 1:00 noon Instructons: 1. You have four hours to answer questons n ths examnaton.. You must show your work to receve
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Experment-I MODULE VII LECTURE - 3 ANALYSIS OF COVARIANCE Dr Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur Any scentfc experment s performed
More informationUNIVERSITY OF TORONTO Faculty of Arts and Science. December 2005 Examinations STA437H1F/STA1005HF. Duration - 3 hours
UNIVERSITY OF TORONTO Faculty of Arts and Scence December 005 Examnatons STA47HF/STA005HF Duraton - hours AIDS ALLOWED: (to be suppled by the student) Non-programmable calculator One handwrtten 8.5'' x
More informationChapter 8 Indicator Variables
Chapter 8 Indcator Varables In general, e explanatory varables n any regresson analyss are assumed to be quanttatve n nature. For example, e varables lke temperature, dstance, age etc. are quanttatve n
More informationEcon Statistical Properties of the OLS estimator. Sanjaya DeSilva
Econ 39 - Statstcal Propertes of the OLS estmator Sanjaya DeSlva September, 008 1 Overvew Recall that the true regresson model s Y = β 0 + β 1 X + u (1) Applyng the OLS method to a sample of data, we estmate
More informationQuestion 1 carries a weight of 25%; question 2 carries 20%; question 3 carries 25%; and question 4 carries 30%.
UNIVERSITY OF EAST ANGLIA School of Economcs Man Seres PGT Examnaton 017-18 FINANCIAL ECONOMETRICS ECO-7009A Tme allowed: HOURS Answer ALL FOUR questons. Queston 1 carres a weght of 5%; queston carres
More informationInterval Estimation in the Classical Normal Linear Regression Model. 1. Introduction
ECONOMICS 35* -- NOTE 7 ECON 35* -- NOTE 7 Interval Estmaton n the Classcal Normal Lnear Regresson Model Ths note outlnes the basc elements of nterval estmaton n the Classcal Normal Lnear Regresson Model
More informationLecture 9: Linear regression: centering, hypothesis testing, multiple covariates, and confounding
Lecture 9: Lnear regresson: centerng, hypothess testng, multple covarates, and confoundng Sandy Eckel seckel@jhsph.edu 6 May 008 Recall: man dea of lnear regresson Lnear regresson can be used to study
More informationProfessor Chris Murray. Midterm Exam
Econ 7 Econometrcs Sprng 4 Professor Chrs Murray McElhnney D cjmurray@uh.edu Mdterm Exam Wrte your answers on one sde of the blank whte paper that I have gven you.. Do not wrte your answers on ths exam.
More informationECON 351* -- Note 23: Tests for Coefficient Differences: Examples Introduction. Sample data: A random sample of 534 paid employees.
Model and Data ECON 35* -- NOTE 3 Tests for Coeffcent Dfferences: Examples. Introducton Sample data: A random sample of 534 pad employees. Varable defntons: W hourly wage rate of employee ; lnw the natural
More informationLecture 9: Linear regression: centering, hypothesis testing, multiple covariates, and confounding
Recall: man dea of lnear regresson Lecture 9: Lnear regresson: centerng, hypothess testng, multple covarates, and confoundng Sandy Eckel seckel@jhsph.edu 6 May 8 Lnear regresson can be used to study an
More informationComposite Hypotheses testing
Composte ypotheses testng In many hypothess testng problems there are many possble dstrbutons that can occur under each of the hypotheses. The output of the source s a set of parameters (ponts n a parameter
More informationNow we relax this assumption and allow that the error variance depends on the independent variables, i.e., heteroskedasticity
ECON 48 / WH Hong Heteroskedastcty. Consequences of Heteroskedastcty for OLS Assumpton MLR. 5: Homoskedastcty var ( u x ) = σ Now we relax ths assumpton and allow that the error varance depends on the
More information28. SIMPLE LINEAR REGRESSION III
8. SIMPLE LINEAR REGRESSION III Ftted Values and Resduals US Domestc Beers: Calores vs. % Alcohol To each observed x, there corresponds a y-value on the ftted lne, y ˆ = βˆ + βˆ x. The are called ftted
More informationLecture 6 More on Complete Randomized Block Design (RBD)
Lecture 6 More on Complete Randomzed Block Desgn (RBD) Multple test Multple test The multple comparsons or multple testng problem occurs when one consders a set of statstcal nferences smultaneously. For
More informationLearning Objectives for Chapter 11
Chapter : Lnear Regresson and Correlaton Methods Hldebrand, Ott and Gray Basc Statstcal Ideas for Managers Second Edton Learnng Objectves for Chapter Usng the scatterplot n regresson analyss Usng the method
More informationSoc 3811 Basic Social Statistics Third Midterm Exam Spring 2010
Soc 3811 Basc Socal Statstcs Thrd Mdterm Exam Sprng 2010 Your Name [50 ponts]: ID #: Your TA: Kyungmn Baek Meghan Zacher Frank Zhang INSTRUCTIONS: (A) Wrte your name on the lne at top front of every sheet.
More informationLinear regression. Regression Models. Chapter 11 Student Lecture Notes Regression Analysis is the
Chapter 11 Student Lecture Notes 11-1 Lnear regresson Wenl lu Dept. Health statstcs School of publc health Tanjn medcal unversty 1 Regresson Models 1. Answer What Is the Relatonshp Between the Varables?.
More informationEcon107 Applied Econometrics Topic 9: Heteroskedasticity (Studenmund, Chapter 10)
I. Defnton and Problems Econ7 Appled Econometrcs Topc 9: Heteroskedastcty (Studenmund, Chapter ) We now relax another classcal assumpton. Ths s a problem that arses often wth cross sectons of ndvduals,
More informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 31 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 6. Rdge regresson The OLSE s the best lnear unbased
More informationProperties of Least Squares
Week 3 3.1 Smple Lnear Regresson Model 3. Propertes of Least Squares Estmators Y Y β 1 + β X + u weekly famly expendtures X weekly famly ncome For a gven level of x, the expected level of food expendtures
More informationChapter 9: Statistical Inference and the Relationship between Two Variables
Chapter 9: Statstcal Inference and the Relatonshp between Two Varables Key Words The Regresson Model The Sample Regresson Equaton The Pearson Correlaton Coeffcent Learnng Outcomes After studyng ths chapter,
More informationAnswers Problem Set 2 Chem 314A Williamsen Spring 2000
Answers Problem Set Chem 314A Wllamsen Sprng 000 1) Gve me the followng crtcal values from the statstcal tables. a) z-statstc,-sded test, 99.7% confdence lmt ±3 b) t-statstc (Case I), 1-sded test, 95%
More informationChapter 2 - The Simple Linear Regression Model S =0. e i is a random error. S β2 β. This is a minimization problem. Solution is a calculus exercise.
Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where y + = β + β e for =,..., y and are observable varables e s a random error How can an estmaton rule be constructed for the
More informationLecture 4 Hypothesis Testing
Lecture 4 Hypothess Testng We may wsh to test pror hypotheses about the coeffcents we estmate. We can use the estmates to test whether the data rejects our hypothess. An example mght be that we wsh to
More informationSTAT 405 BIOSTATISTICS (Fall 2016) Handout 15 Introduction to Logistic Regression
STAT 45 BIOSTATISTICS (Fall 26) Handout 5 Introducton to Logstc Regresson Ths handout covers materal found n Secton 3.7 of your text. You may also want to revew regresson technques n Chapter. In ths handout,
More informationChapter 8 Multivariate Regression Analysis
Chapter 8 Multvarate Regresson Analyss 8.3 Multple Regresson wth K Independent Varables 8.4 Sgnfcance tests of Parameters Populaton Regresson Model For K ndependent varables, the populaton regresson and
More informationChapter 14 Simple Linear Regression Page 1. Introduction to regression analysis 14-2
Chapter 4 Smple Lnear Regresson Page. Introducton to regresson analyss 4- The Regresson Equaton. Lnear Functons 4-4 3. Estmaton and nterpretaton of model parameters 4-6 4. Inference on the model parameters
More informationTesting for seasonal unit roots in heterogeneous panels
Testng for seasonal unt roots n heterogeneous panels Jesus Otero * Facultad de Economía Unversdad del Rosaro, Colomba Jeremy Smth Department of Economcs Unversty of arwck Monca Gulett Aston Busness School
More informationwhere I = (n x n) diagonal identity matrix with diagonal elements = 1 and off-diagonal elements = 0; and σ 2 e = variance of (Y X).
11.4.1 Estmaton of Multple Regresson Coeffcents In multple lnear regresson, we essentally solve n equatons for the p unnown parameters. hus n must e equal to or greater than p and n practce n should e
More informationChapter 12 Analysis of Covariance
Chapter Analyss of Covarance Any scentfc experment s performed to know somethng that s unknown about a group of treatments and to test certan hypothess about the correspondng treatment effect When varablty
More informationContinuous vs. Discrete Goods
CE 651 Transportaton Economcs Charsma Choudhury Lecture 3-4 Analyss of Demand Contnuous vs. Dscrete Goods Contnuous Goods Dscrete Goods x auto 1 Indfference u curves 3 u u 1 x 1 0 1 bus Outlne Data Modelng
More informationβ0 + β1xi. You are interested in estimating the unknown parameters β
Ordnary Least Squares (OLS): Smple Lnear Regresson (SLR) Analytcs The SLR Setup Sample Statstcs Ordnary Least Squares (OLS): FOCs and SOCs Back to OLS and Sample Statstcs Predctons (and Resduals) wth OLS
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More information/ n ) are compared. The logic is: if the two
STAT C141, Sprng 2005 Lecture 13 Two sample tests One sample tests: examples of goodness of ft tests, where we are testng whether our data supports predctons. Two sample tests: called as tests of ndependence
More informatione i is a random error
Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where + β + β e for,..., and are observable varables e s a random error How can an estmaton rule be constructed for the unknown
More informationCHAPTER 8 SOLUTIONS TO PROBLEMS
CHAPTER 8 SOLUTIONS TO PROBLEMS 8.1 Parts () and (). The homoskedastcty assumpton played no role n Chapter 5 n showng that OLS s consstent. But we know that heteroskedastcty causes statstcal nference based
More informationTopic 7: Analysis of Variance
Topc 7: Analyss of Varance Outlne Parttonng sums of squares Breakdown the degrees of freedom Expected mean squares (EMS) F test ANOVA table General lnear test Pearson Correlaton / R 2 Analyss of Varance
More informationPredictive Analytics : QM901.1x Prof U Dinesh Kumar, IIMB. All Rights Reserved, Indian Institute of Management Bangalore
Sesson Outlne Introducton to classfcaton problems and dscrete choce models. Introducton to Logstcs Regresson. Logstc functon and Logt functon. Maxmum Lkelhood Estmator (MLE) for estmaton of LR parameters.
More informationSee Book Chapter 11 2 nd Edition (Chapter 10 1 st Edition)
Count Data Models See Book Chapter 11 2 nd Edton (Chapter 10 1 st Edton) Count data consst of non-negatve nteger values Examples: number of drver route changes per week, the number of trp departure changes
More informationANOVA. The Observations y ij
ANOVA Stands for ANalyss Of VArance But t s a test of dfferences n means The dea: The Observatons y j Treatment group = 1 = 2 = k y 11 y 21 y k,1 y 12 y 22 y k,2 y 1, n1 y 2, n2 y k, nk means: m 1 m 2
More informationTests of Exclusion Restrictions on Regression Coefficients: Formulation and Interpretation
ECONOMICS 5* -- NOTE 6 ECON 5* -- NOTE 6 Tests of Excluson Restrctons on Regresson Coeffcents: Formulaton and Interpretaton The populaton regresson equaton (PRE) for the general multple lnear regresson
More informationSTAT 511 FINAL EXAM NAME Spring 2001
STAT 5 FINAL EXAM NAME Sprng Instructons: Ths s a closed book exam. No notes or books are allowed. ou may use a calculator but you are not allowed to store notes or formulas n the calculator. Please wrte
More informationBasically, if you have a dummy dependent variable you will be estimating a probability.
ECON 497: Lecture Notes 13 Page 1 of 1 Metropoltan State Unversty ECON 497: Research and Forecastng Lecture Notes 13 Dummy Dependent Varable Technques Studenmund Chapter 13 Bascally, f you have a dummy
More informationIntroduction to Dummy Variable Regressors. 1. An Example of Dummy Variable Regressors
ECONOMICS 5* -- Introducton to Dummy Varable Regressors ECON 5* -- Introducton to NOTE Introducton to Dummy Varable Regressors. An Example of Dummy Varable Regressors A model of North Amercan car prces
More information4 Analysis of Variance (ANOVA) 5 ANOVA. 5.1 Introduction. 5.2 Fixed Effects ANOVA
4 Analyss of Varance (ANOVA) 5 ANOVA 51 Introducton ANOVA ANOVA s a way to estmate and test the means of multple populatons We wll start wth one-way ANOVA If the populatons ncluded n the study are selected
More informationStatistics MINITAB - Lab 2
Statstcs 20080 MINITAB - Lab 2 1. Smple Lnear Regresson In smple lnear regresson we attempt to model a lnear relatonshp between two varables wth a straght lne and make statstcal nferences concernng that
More information( )( ) [ ] [ ] ( ) 1 = [ ] = ( ) 1. H = X X X X is called the hat matrix ( it puts the hats on the Y s) and is of order n n H = X X X X.
( ) ( ) where ( ) 1 ˆ β = X X X X β + ε = β + Aε A = X X 1 X [ ] E ˆ β β AE ε β so ˆ = + = β s unbased ( )( ) [ ] ˆ Cov β = E ˆ β β ˆ β β = E Aεε A AE ε ε A Aσ IA = σ AA = σ X X = [ ] = ( ) 1 Ftted values
More information