Linear Regression Analysis: Terminology and Notation
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1 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 by the followng populaton regresson equaton (called the PRE for short): Y f (X ) + u β + β X + u The PRF (populaton regresson functon): f (X ) β + βx the -th value of the populaton regresson functon (PRF). Observable Varables: Y the -th value of the dependent varable Y X the -th value of the ndependent varable X Unobservable Varable: u the random error term for the -th member of the populaton Unknown Parameters: the regresson coeffcents β the ntercept coeffcent β the slope coeffcent on X The true populaton values of the regresson coeffcents β and β are unknown. ECON 35* -- Secton : Flename 35lecture2.doc... Page of 25
2 ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page 2) Varables and Parameters PRE: Y f (X ) + u β + β X + u The varables of the regresson model are Y, X, and u. Y and X are the observable varables; ther values can be observed or measured. Y s called any of the followng: X s called any of the followng: () the dependent varable (2) the regressand (3) the explaned varable. () the ndependent varable (2) the regressor (3) the explanatory varable. u s an unobservable random varable; ts value cannot be observed or measured. It s called a random error term. β and β are the parameters of the regresson model, together wth any unknown parameters of the probablty dstrbuton of the random error term u. β and β are called regresson coeffcents; n partcular, and β the ntercept coeffcent, β the slope coeffcent of X. The true populaton values of the regresson coeffcents β and β are unknown. ECON 35* -- Secton : Flename 35lecture2.doc... Page 2 of 25
3 ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page 3) Smple Regresson versus Multple Regresson A smple regresson model has only two observable varables: () one dependent varable or regressand Y ; (2) one ndependent varable or regressor X. Populaton regresson equaton or PRE for a smple lnear regresson model: Y β + β X + u A multple regresson model has three or more observable varables: () one dependent varable or regressand Y ; (2) two or more ndependent varables or regressors X, X 2,..., X k, where X j the -th value of the j-th regressor X j (j, 2,, k). Populaton regresson equaton or PRE for a multple lnear regresson model: Y β + β X + β X + L + β X k k u ECON 35* -- Secton : Flename 35lecture2.doc... Page 3 of 25
4 ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page 4) The Smple Lnear Regresson Model The PRE (populaton regresson equaton) for the smple lnear regresson model: Y f (X ) + u β + β X + u (a) PRF random error f (X ) β + βx the PRF (populaton regresson functon) for the -th populaton member u Y f (X ) Y β βx the random error for the -th populaton member β, β the unknown regresson coeffcents β and β Number of regresson coeffcents K 2. Number of slope coeffcents K 2. Sample Data: A random sample of N members of the populaton for whch the observed values of Y and X are measured. Each sample observaton s of the form (Y, X ),,..., N The SRE (sample regresson equaton) for the smple lnear regresson model: Y fˆ(x ) + û Ŷ + û βˆ + βˆ X + û (b) SRF resdual fˆ (X ) Ŷ ˆ ˆ β + βx the SRF (sample regresson functon) for sample observaton û Y fˆ(x ) Y ˆ ˆ Ŷ Y β βx the resdual for sample observaton βˆ, βˆ estmators or estmates of the regresson coeffcents β and β ECON 35* -- Secton : Flename 35lecture2.doc... Page 4 of 25
5 ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page 5) The Multple Lnear Regresson Model The PRE (populaton regresson equaton) for the multple lnear regresson model s: Y f (X, X, K, X ) + u β + β X + β X + L+ β X + u (2a) f (X u 2 k 2 PRF, X2, K, Xk) β + βx + β2x2 + L+ βkxk 2 k k random error the PRF (populaton regresson functon) for the -th populaton member Y f (X, X 2, K, X k ) Y β βx β2x 2 L β the random error for the -th populaton member β, K β the unknown regresson coeffcents β, β, β 2,, β k β, β2,, k Number of regresson coeffcents K. Number of slope coeffcents k K. Sample Data: A random sample of N members of the populaton for whch the observed values of Y and X, X 2,, X k are measured. Each sample observaton s of the form (Y, X, X 2,, X k ),,..., N The SRE (sample regresson equaton) for the multple lnear regresson model: Y fˆ(x, X, K, X ) + û Ŷ + û βˆ + βˆ X + βˆ X + L+ βˆ X + û (2b) fˆ (X û 2 k ˆ 2 SRF, X 2, K, X k ) Ŷ β + βx + β2x 2 + L+ βkx k ˆ the SRF (sample regresson functon) for sample observaton Y fˆ(x, X 2, K, X ˆ ˆ ˆ ˆ k ) Y Ŷ Y β βx β2x 2 L β the resdual for sample observaton βˆ, βˆ, βˆ, K, βˆ estmators or estmates of the regresson coeffcents 2 k ˆ ˆ 2 k X k k k resdual k X k ECON 35* -- Secton : Flename 35lecture2.doc... Page 5 of 25
6 ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page 6) Examples of multple regresson models A three-varable lnear regresson model has two regressors; ts PRE s wrtten as Y β + β X + β X + u. 2 2 Total number of regresson coeffcents K 3 Number of slope coeffcents k K 3 2 A four-varable lnear regresson model has three regressors; ts PRE s wrtten as Y β + β X + β X + β X + u Total number of regresson coeffcents K 4 Number of slope coeffcents k K 4 3 The general multple lnear regresson model has k K regressors; ts PRE s wrtten as Y β + β X + β X + L + β X + u. 2 2 k k Total number of regresson coeffcents K. Number of slope coeffcents k K. ECON 35* -- Secton : Flename 35lecture2.doc... Page 6 of 25
7 ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page 7) Regresson Analyss: A Hypothetcal Numercal Example Reference: D. Gujarat (995), Chapter 2, pp Purpose: To llustrate some of the basc deas of lnear regresson analyss. The Model: A smple consumpton functon representng the relatonshp between Y the weekly consumpton expendture of famly ($ per week); X the weekly dsposable (after-tax) ncome of famly ($ per week); The PRE (populaton regresson equaton) for ths model can be wrtten as Y β + β X + u () The Populaton: conssts entrely of 6 famles. We assume that the weekly dsposable ncomes of these famles take only dstnct values --.e., X takes only the dstnct values X 8,, 2, 4, 6, 8, 2, 22, 24, 26. We further assume that we can observe the entre populaton of 6 famles. The data for the complete populaton s gven n Table 2.. ECON 35* -- Secton : Flename 35lecture2.doc... Page 7 of 25
8 ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page 8) Table 2.: Populaton data ponts (Y, X ) for the populaton of 6 famles. X values Y values Sum Y values Number of Y Interpretaton of Table 2.: Each column of Table 2. represents the populaton condtonal dstrbuton of Y (famles weekly consumpton expendture) for the correspondng value of X (famles weekly dsposable ncome). The frst column gves the condtonal dstrbuton of Y for X 8; fve famles n the populaton have weekly dsposable ncome equal to 8 dollars. The ffth column gves the condtonal dstrbuton of Y for X 6; sx famles n the populaton have weekly dsposable ncome equal to 6 dollars. The tenth (last) column gves the condtonal dstrbuton of Y for X 26; seven famles n the populaton have weekly dsposable ncome equal to 26 dollars. ECON 35* -- Secton : Flename 35lecture2.doc... Page 8 of 25
9 ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page 9) Table 2.2: Populaton condtonal probabltes of Y for each populaton value of X. Notaton: ( Y X ) p( Y X ) p the condtonal probablty of Y for X X j the probablty that the random varable Y takes the numercal value Y j gven that the varable X s equal to the numercal value X. Condtonal probabltes ( Y ) p for the populaton data n Table 2. X X values ( Y ) p X /5 /6 /5 /7 /6 /6 /5 /7 /6 /7 /5 /6 /5 /7 /6 /6 /5 /7 /6 /7 /5 /6 /5 /7 /6 /6 /5 /7 /6 /7 /5 /6 /5 /7 /6 /6 /5 /7 /6 /7 /5 /6 /5 /7 /6 /6 /5 /7 /6 /7 -- /6 -- /7 /6 /6 -- /7 /6 / / /7 -- /7 Sum Y values Number of Y ( Y ) E X Interpretaton of Table 2.2: Each column of Table 2.2 contans the populaton condtonal probabltes of Y (famles weekly consumpton expendture) for the correspondng value of X (famles weekly dsposable ncome). ECON 35* -- Secton : Flename 35lecture2.doc... Page 9 of 25
10 ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page ) Examples: Computng the Condtonal Probabltes of Indvdual Y Values. Consder the column correspondng to X 8. There are fve dfferent values of Y for X 8: Y X 8: 55, 6, 65, 7, 75. The probablty of observng any one famly whose weekly dsposable ncome s X 8 equals /5: e.g., p(y 55 X 8) 5. p(y 6 X 8) 5. p(y 65 X 8) 5. p(y 7 X 8) 5. p(y 75 X 8) Consder the column correspondng to X 6. There are sx dfferent values of Y for X 6: Y X 6: 2, 7,, 6, 8, 25. The probablty of observng any one famly whose weekly dsposable ncome s X 6 equals /6: e.g., p(y 2 X 6) 6. p(y X 6) 6. ECON 35* -- Secton : Flename 35lecture2.doc... Page of 25
11 ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page ) Populaton Condtonal Means of Y For each of the populaton values of X, we can compute from Tables 2. and 2.2 the correspondng condtonal mean value of the populaton values of Y. For each of the values X of X, the populaton mean value of Y s called or () the populaton condtonal mean of Y (2) the populaton condtonal expectaton of Y. Notaton: ( Y X ) E( Y X ) E Defnton: E X the populaton condtonal mean of Y for X X the expected value of Y gven that X takes the specfc value X " ( Y X ) E( Y X X ) p( Y X ) where ( Y ) X X X p the condtonal probablty of Y when X X ; p( Y X )Y the product of each populaton value of Y and ts correspondng condtonal probablty for X X. In words, the above formula for E ( Y X ) E( Y X ) X of X, Y says that for the value X () multply each populaton value of Y by ts assocated condtonal probablty p Y to get the product ( Y X )Y ( ) X p (2) then sum these products p( Y X )Y over all the populaton values of Y correspondng to X X. ECON 35* -- Secton : Flename 35lecture2.doc... Page of 25
12 ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page 2) Illustratve Calculatons of ( Y ). For X 8, ( Y ) p /5: X E : X E ( Y X 8) For X 6, ( Y ) p /6: X E ( Y X 6) For X 26, ( Y ) p /7: X E ( Y X 26) ECON 35* -- Secton : Flename 35lecture2.doc... Page 2 of 25
13 ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page 3) Table 2.3: Populaton Condtonal Means of Y Table 2.3 X E ( Y ) X Interpretaton of Table 2.3: Table 2.3 tabulates the relatonshp between ( Y ) populaton of 6 famles. Ths populaton relatonshp between ( Y ) X E and for ths partcular X X X E and s called ether or () the populaton regresson functon, or PRF. (2) the populaton condtonal mean functon, or populaton CMF So Table 2.3 s a tabular representaton of the PRF for the populaton of 6 famles. ECON 35* -- Secton : Flename 35lecture2.doc... Page 3 of 25
14 ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page 4) Propertes of the Populaton Regresson Functon, or PRF: Table 2.3 X E ( Y ) E ( Y ) s a functon of X :.e., E( Y X ) f (X ) X 2. ( Y ) E s an ncreasng functon of X :.e., X X. X > E( Y X ) > and X < 3. ( Y ) E s a lnear functon of X :.e., X E( Y X ) < A plot of the ponts n Table 2.3 le on a straght lne.. Each 2-dollar ncrease n X nduces a constant 2-dollar ncrease n E Y :.e., ( ) X X 2 E( Y X ) 2 E ( Y X ) X ECON 35* -- Secton : Flename 35lecture2.doc... Page 4 of 25
15 ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page 5) 4. The populaton regresson functon (PRF) -- also called the populaton condtonal mean functon -- takes the general lnear form E ( Y X ) β + βx. 5. The populaton values of the regresson coeffcents β and β 2 for ths hypothetcal populaton of 6 famles are: β 7 and β The populaton regresson functon, or PRF, for ths partcular populaton of 6 famles s therefore ( Y X ) β + βx 7.6 X E +. Summary -- The Populaton Regresson Functon (PRF) The PRF, or populaton regresson functon, for ths hypothetcal populaton of 6 famles s a lnear functon of the populaton values X of the regressor X; t takes the form where ( Y X ) β + βx 7.6 X f (X ) E +. β 7 s the populaton value of the ntercept coeffcent β.6 s the populaton value of the slope coeffcent of X. ECON 35* -- Secton : Flename 35lecture2.doc... Page 5 of 25
16 ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page 6) Fgure 2. Plot of Populaton Data Ponts, Condtonal Means E(Y X), and the Populaton Regresson Functon PRF Y Ftted values 2 Weekly consumpton expendture, $ PRF E(Y X) Weekly ncome, $. The small dots n Fgure 2. consttute a scatterplot of the populaton values of Y and X for the populaton of 6 famles: Each small dot corresponds to a sngle populaton data pont of the form (Y, X ), 2,..., The sold lne n Fgure 2. s the populaton regresson lne for the populaton of 6 famles. Each par of populaton values of ( E (Y X ), ) square dot n Fgure 2.. X, s represented by a large Ths populaton regresson lne s the locus of the ponts n Table e., E (Y X ),,,...,. t connects the ponts of the form ( ) X ECON 35* -- Secton : Flename 35lecture2.doc... Page 6 of 25
17 ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page 7) The Random Error Terms Defnton: The unobservable random error term for the -th populaton member s denoted as u and defned as u ( Y X ) Y E. For each populaton member -- for each of the 6 famles n our hypothetcal populaton -- the random error term u equals the devaton of that populaton member's ndvdual Y value from the populaton condtonal mean value of Y for the correspondng value X of X. Termnology: The random error term u s also known as the stochastc error term, the random dsturbance term, or the stochastc dsturbance term Implcaton : By smple re-arrangement of the above defnton of u, t s obvous that each ndvdual populaton value Y of Y can be wrtten as ( Y X ) u β + β X + E( Y X ) β + βx Y E + snce u. Ths equaton s called the populaton regresson equaton, or PRE. Interpretaton: The PRE ndcates that each populaton value Y of Y can be expressed as the sum of two components: E Y X β + β X () ( ) the populaton condtonal mean of Y for X X the mean weekly consumpton expendture for all famles n the populaton who have weekly dsposable ncome X X. (2) u the random error term for the -th populaton member Y E Y ( ) X the devaton of famly s weekly consumpton expendture Y from the populaton mean value E(Y X ) of all famles n the populaton that have the same weekly dsposable ncome X X. ECON 35* -- Secton : Flename 35lecture2.doc... Page 7 of 25
18 ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page 8) Implcaton 2: The populaton condtonal mean value of the random error terms for each populaton value X of X equals --.e., Proof: E ( u X ).. Take the condtonal expectaton for X X of both sdes of the PRE: E ( Y X ) E[ E( Y X )] + E( u X ) E( Y X ) + E( u X ) snce E( Y X ) s a constant. 2. Snce EY ( X) EY ( X), the above equaton mples that Eu ( X). What do the Random Error Terms u Represent? The random error terms represent all the unknown and unobservable varables other than X that determne the ndvdual populaton values Y of the dependent varable Y. They arse from the followng factors:. Omtted varables that determne the populaton Y values 2. Intrnsc randomness n ndvdual behavour ECON 35* -- Secton : Flename 35lecture2.doc... Page 8 of 25
19 ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page 9) Random Errors for Hypothetcal Populaton of 6 Famles Random Error Terms for X Y E(Y X ) u Y E(Y X ) Sum 462 Sum Mean 462/6 77 Mean Random Error Terms for X 8 Y E(Y X 8) u Y E(Y X 8) Sum 75 Sum Mean 75/6 25 Mean Random Error Terms for X 24 Y E(Y X 24) u Y E(Y X 24) Sum 966 Sum Mean 966/6 6 Mean ECON 35* -- Secton : Flename 35lecture2.doc... Page 9 of 25
20 ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page 2) The Sample Regresson Functon Important Pont : Snce n practce we do not observe the entre relevant populaton, and never know the true PRF, we must estmate the PRF from sample data. Objectve of Regresson Analyss: To estmate the PRF (populaton regresson functon) from sample data consstng of N randomly selected observatons (Y, X ),,..., N taken from the populaton. Form of the Sample Regresson Functon (SRF): The sample regresson functon, or SRF, takes the general form Ŷ β ˆ + β ˆ X (,..., N) where $Y an estmate of the PRF, f (X ) E(Y X ) β + β X ; ˆβ an estmate of the ntercept coeffcent β ; $β an estmate of the slope coeffcent β. Nature of the Sample Data: A sample s a randomly-selected subset of populaton members.. The sample observatons {(Y, X ):,..., N} are typcally a small subset of the parent populaton of all populaton data ponts (Y, X ). Sample sze N s much smaller than the number of populaton data ponts. 2. Each random sample from a gven populaton yelds one estmate of the PRF --.e., one estmate of the numercal value of β, and one estmate of the numercal value of β. ECON 35* -- Secton : Flename 35lecture2.doc... Page 2 of 25
21 ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page 2) Important Pont 2: Each random sample from the same populaton yelds a dfferent SRF --.e., a dfferent numercal value of ˆβ, and a dfferent numercal value of $β. Example: Consder two random samples of observatons from the populaton of 6 famles. Each sample conssts of one famly for each of the dfferent populaton values of X. X Tables 2.4 and 2.5 Sample Sample 2 Y X Y Because the two samples contan dfferent Y values for the X values, they wll yeld dfferent SRFs -- a dfferent numercal value of ˆβ, and a dfferent numercal value of $β. Sample SRF (SRF ): Y$ X, where the Sample coeffcent estmates are β ˆ () and βˆ ( ).59 Sample 2 SRF (SRF 2 ): Y$ X, where the Sample 2 coeffcent estmates are β ˆ (2) 7.7 and βˆ (2 ).576 ECON 35* -- Secton : Flename 35lecture2.doc... Page 2 of 25
22 ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page 22) Fgure 2.2 Plot of Sample Data Ponts and Sample Regresson Functons for Random Samples and 2 SRF s the SRF based on Sample : Y$ X SRF 2 s the SRF based on Sample 2: Y$ X SRF s the flatter regresson lne, SRF 2 s the steeper regresson lne. Important Ponts: () Nether of these SRFs s dentcal to the true PRF. Each s merely an approxmaton to the true PRF. (2) How good an approxmaton any SRF provdes to the true PRF depends on how the SRF s constructed from sample data --.e., on the propertes of the coeffcent estmators ˆβ and β $. Y Y2 2 Weekly consumpton expendture, $ Weekly ncome, $ ECON 35* -- Secton : Flename 35lecture2.doc... Page 22 of 25
23 ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page 23) The Sample Regresson Equaton (SRE) The sample regresson equaton (SRE) s the sample counterpart of the populaton regresson equaton (PRE) Y f (X ) + u E(Y X ) + u β + β X + u the PRE Form of the Sample Regresson Equaton (SRE): The sample regresson equaton, or SRE, takes the general form Y Ŷ + û βˆ + βˆ X + û (,..., N) the SRE where Ŷ β ˆ + β ˆ X an estmate of the PRF, f (X ) E(Y X ) β + βx ; ˆβ an estmate of the ntercept coeffcent β ; $β an estmate of the slope coeffcent β. $u the resdual for sample observaton. Interpretaton of the SRE: The SRE represents each sample value of Y -- each Y value -- as the sum of two components: () the estmated (or predcted) mean value of Y for each sample value X of X,.e., Ŷ β ˆ + β ˆ X (,..., N); (2) the resdual correspondng to the -th sample observaton,.e., û Y Ŷ Y β ˆ β ˆ X (,..., N). û the resdual for the -th sample observaton the observed Y-value ( Y ) the estmated Y-value ( Ŷ ) ECON 35* -- Secton : Flename 35lecture2.doc... Page 23 of 25
24 ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page 24) Compare the Populaton and Sample Regresson Equatons: the PRE and SRE The PRE for Y s: Y f (X ) + u E(Y X ) + u β + β X + u The SRE for Y s: Y Ŷ + û βˆ + βˆ X + û Fgure 2.3: Comparson of Populaton and Sample Regresson Lnes Y Y (Y, X ) SRF Ŷ PRF E(Y X ) X X The populaton regresson lne s a plot of the PRF: E(Y X ) β + βx. The sample regresson lne s a plot of the SRF: Ŷ β ˆ + β ˆ X. ECON 35* -- Secton : Flename 35lecture2.doc... Page 24 of 25
25 ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page 25) Fgure 2.3: Comparson of Populaton and Sample Regresson Lnes Y Y (Y, X ) SRF Ŷ PRF E(Y X ) At X X : X X The populaton regresson equaton (PRE) represents the populaton value Y of Y as the sum of two parts: Y + u E(Y X ) + u β + βx u, where E(Y X ) β + βx Y E(Y X ) Y β β X dstance between Y and E(Y X ) The sample regresson equaton (SRE) represents the populaton value Y of Y as the sum of two parts: Y ˆ ˆ + û Ŷ + û β + βx û, where Ŷ β + βx Y Ŷ Y β ˆ β ˆ X dstance between Y and Ŷ. ˆ ˆ ECON 35* -- Secton : Flename 35lecture2.doc... Page 25 of 25
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