Handbook Limdep

LIMDEP :

2550

LIMDEP : ii

ii 1. 1 2. LIMDEP 1

2.1 LIMDEP 2 2.2 Main Menus 4 2.3 Tool Bar 5 2.4 Command Bar 5 2.5 Options LIMDEP 8 2.6 Project 10 2.7 LIMDEP 14

3. LIMDEP 15 3.1 18 3.2 Project 21 3.3 Transform 23 3.4 26 Historgram Plot Variable Multiple Scatter Plots

3.5 33 4. (Ordinary Least Square Method) 36

4.1 36 4.2 37 4.3 LIMDEP 38 4.4 LIMDEP Stepwise 43 4.5 47 4.6 50

5. Multicollinearity 54 5.1 Multicollinearity 54 5.2 Multicollinearity 54 Simple Correlation Coefficients Variance Inflation Factors (VIF)

5.3 Multicollinearity 57

LIMDEP : iii

6. Heteroskedasticity 57

6.1 Heteroskedasticity 57 6.2 Heteroskedasticity 58 6.3 Heteroskedasticity 59 Heteroskedasticity-Corrected Standard Erros Weighted Least Square (WLS)

7. Autocorrelation 63 7.1 Autocorrelation 63 7.2 Autocorrelation 64 7.3 Autocorrelation 66

8. Binary Choice Models 70 8.1 Probit 73 8.2 Logit 79

9. Ordered Choice Models 85 9.1 Ordered Probit 86 9.2 Ordered Logit 92

10. Multinomial Logit Models 94 10.1 The Multinomail Logit Models 95 10.2 The Coditional Logit Models 101

11. Poisson Regression Models 110 12. Tobit Models 117 13. Sample Selection Models 124 14. Switching Regressions Models 135 15. Stochastic Frontiers Models 143 150

LIMDEP :

1.

EViews, LIMDEP, SHAZAM, STATA LIMDEP Stochastic Frontier, Switching Regression, Tobit Models LIMDEP LIMited DEPendent variable modeling .. 2523 Tobit Econometric Software LIMDEP Version 7.0 LIMDEP 2 LIMDEP NLOGIT (.. 2550) Econometric Software LIMDEP Version 9.0 NLOGIT Version 4.0 Count Data, Panel Data Stochastic Frontier Anlysis [SFA] Data Envelopment Analysis [DEA] LIMDEP LIMDEP Cross section Panel data

2. LIMDEP

LIMDEP Project LIMDEP SHAZAM, STATA RATS 3 (Data Editor) (Command Document) (Output) 3 Wizards Command LIMDEP LIMDEP NLOGIT Version 3.0 [LIMDEP 8.0]

LIMDEP : 2

2.1 LIMDEP

LIMDEP

LIMDEP Microsoft Window feature Main menu, Minimize window, Maximize window, Closed Pull down menu Microsoft

Double-click

Restore/Maximize

Minimize Close

Project window

Tool Bar

Command Bar Function Button

22,222 Rows 22,222 Obs.

Status Bar

Main Menus

LIMDEP : 3

Tip LIMDEP

Mouse Double click [ Project] Tip Mouse

LIMDEP ESC / LIMDEP ESC

LIMDEP ALT F4 3 LIMDEP Data Editor, Command Document Output

[] Scroll Bar Microsoft

/

LIMDEP : 4

LIMDEP File/Exit Double click Limdep button Alt F4 LIMDEP LIMDEP Save

Save Save LIMDEP LIMDEP

LIMDEP

2.2 Main Menus LIMDEP

LIMDEP 9

Scroll Arrow

Horizontal Scroll Bar Scroll Buttons

Scroll Arrow

Vertical Scroll Bar

Split Box

Split Box

Scroll Buttons

LIMDEP : 5

File Menu: Project files Command files Edit Menu: Edit Cut, Copy, Paste, Clear, Delete Insert Menu: Project Menu: Project Set Sample Model Menu: Wizeards Linear Models Run Menu: Command Command Document Tools Menu: Scalar/Matrics Tables

Options Limdep Window Menu: Desktop Help Menu: LIMDEP Help

2.3 Tool Bar

LIMDEP LIMDEP 14

2.4 Command Bar

Command Bar LIMDEP LIMDEP

File

Open

Save

Print

Cut

Copy

Paste

Insert

Run

Stop Run

Pause Run

Project window

Data editor

Output window

LIMDEP : 6

Function Button [ ] LIMDEP Insert Command

Command Bar Enter

LIMDEP [

REGRESS]

LIMDEP : 7

Command Bar Enter LIMDEP

LIMDEP : 8

2.5 Options LIMDEP

Options LIMDEP Tools/Options LIMDEP Options

Options Options Options 5

. View: Options LIMDEP / 3 Tool Bar, Status Bar Command Bar

. Editor: Options Font LIMDEP LIMDEP Font Defualt Courier: 9 Point Font

LIMDEP Font Font LIMDEP Font

Font Options LIMDEP / Automatic Word Editor Tool Bar

LIMDEP : 9

. Projects: Options Default Data Area LIMDEP Data Cells Data Cells Data Cells: 2000000 [2222 Rows 2222 Obs] 15.2588 MB Default Data Area Project Data Area Project Data Area Project Settings Main Menus Project Settings Data Area Project Data Area

Project Settings Data Area Tools Options Default Data Area

. Execution: Options LIMDEP LIMDEP LIMDEP Default Output Message Dialog Boxes

LIMDEP : 10

. Trace: Options LIMDEP (Save) trace LIMDEP Default trace Directory LIMDEP

Options LIMDEP

2.6 Project

LIMDEP LIMDEP Project Defualt Untiled Project window Topic 4 Topic . Data Variable, Namelists, Matrices Scalars . Strings 3 character strings . Procedures 10 Procedures 1 Prucedures . Output Output Window Model Table

LIMDEP : 11

Topic Topic Topic Topic Topic LIMDEP Topic Default Topic

. Data: Topic Default Matrices Scalars Variables Namelist Default [Variables] Namelist Matrices Scalars Default

Matrices: LIMDEP Matrices Default 3 Matrices B, VARB SIGMA Matrices Double click Matrices LIMDEP Matrics LIMDEP Matrices 3 0 Matrix Double click Matirix button [ ] Matrix

Matrix

Double click

LIMDEP : 12

Scalars: LIMDEP Scalars Default 14 Scalars SSQRD, RSQRD, S, SUMSQDEV, RHO, DEGFRDM, SY, YBAR, KREG, NREG, LOGL, LMDA, THETA EXITCODE Scalars Scalars LIMDEP Scalars Project Scalars Double click Scalars LIMDEP New Scalar Scalars New Scalar LIMDEP New Scalar

. Strings: LIMDEP Default Topic Strings 3 Strings Edit

Strings Double click Strings Edit LIMDEP Edit string Edit string Edit string LIMDEP Edit string

Click Scalar

DoubleClick Scalar

LIMDEP : 13

. Output: Topic Default Output Window Tables Default Output Window Double click Output Window LIMDEP Output Output Options Status Trace LIMDEP Default Status

DoubleClick Strings

LIMDEP : 14

2.7 LIMDEP

Directory File Data /File Command /File Output LIMDEP Directory File

LIMDEP YIELD yield Yield

File _ 26 10 digits 8

Ctrl Shift LIMDEP LIMDEP

o One Constant term o b o Varb Matrices o n current sample size o pi 3.14159 o _obsno Observation number current sample size [ CREATE] o _rowno row number data set [ CREATE] o s, sy, ybar, degfrdm, kreg, lmda, logl, nreg, rho, rsqrd, ssqrd, sumsqdev

Scalars o exitcode LIMDEP Procedure

File LIMDEP 2 o Project *.lpj o Command/Output *.lim

Missing data LIMDEP -999 Command document $

LIMDEP $ LIMDEP REGRESS;Lhs=y;Rhs=one,x1,x2$

?

LIMDEP : 15

3. LIMDEP

LIMDEP Command Document LIMDEP Command Command Command Bar Command Command Document Command Document

1 LIMDEP File\New.. Main Menus Ctrl N Tool Bar

LIMDEP : 16

2 New 3 Text/Command Document

LIMDEP Command Document Untitled 1 [ LIMDEP Untiled ]

4 Command Document

LIMDEP Hightlight Tool Bar Run/Run Selection Main Menus Ctrl R LIMDEP Output

LIMDEP : 17

5 Command Document Tool Bar File\Save

Main Menus Ctrl S LIMDEP Save As 6 Directory File

Directory

File

LIMDEP : 18

LIMDEP 1 3 X1 X3, 1 Y 1,000

3.1

LIMDEP Version 8.0 Spreadsheet LIMDEP ASCII File Command Document LIMDEP

READ ; File = [ File ] ; Nobs = [ Observations] ; Nvar = [ Variables] ; Names= [ ,]; ( []) $

Microsoft Excel Microsoft Excel LIMDEP (Row) []

1 LIMDEP Tool Bar LIMDEP Data Editor

LIMDEP : 19

2 LIMDEP Data Editor Mouse Data Editor

3 Import Variables... LIMDEP Import 4 File [ File *.wk *.xls]

LIMDEP Import File Data Editor Data Editor File Microsoft Excel

File

LIMDEP : 20

Output LIMDEP

READ ; FILE = "E:\UN-Other\MY_Paper\Limdep_Intro\Demo.xls" $ [: File ]

Copy Command Bar Command Document 5

Data Editor Data Editor

Data Editor 4 1,000 Observasion

LIMDEP : 21

3.2 Project

Project 1 File/Save Project As Main Menus 2 LIMDEP Save As Directory

Project [ Project *.lpj]

Directory

File

LIMDEP : 22

LIMDEP File Project File Project

1 File/Open Project Main Menus 2 LIMDEP Open Project Directory File

[ Probject *.lpj] LIMDEP File Project

Directory

File

LIMDEP : 23

3.3 Transform

LIMDEP Transform 5 CREATE, DELETE, RECODE, RENAME SORT CREATE [] CREATE

CREATE ; name = expression ; name = expression ; $ CREATE ; X3 = X1+X2 $ [ X3 X1 + X2]

CREATE ; If (logical expression) expression $ CREATE ; If (X1 = 1) | X3 = 3 $ [ X3 X1 = 1 X3 = 3 0] CREATE ; If () name = expression ; (Else) name = expression $ CREATE ; If (age > 21 + job = 1) adult = 1 ; (Else) child = 1$ [ adult child age > 21 job = 1 adult

1 child 0 adult 0 child 0]

Command Bar Command Document Project/New/Variable Main Menus

LIMDEP : 24

LIMDEP New Variable Name Expression Expression LIMDEP Expression Data Editor Tool Bar

CREATE

+ X+Y : a b - XY : a b * X*Y : a b / X/Y : a b ^ X^Y : XY [X Y] @ Box-Cox transformation X@Y : (XY1)/Y logX if Y = 0 a > 0 ! X!Y : max (X,Y) ~ X~Y : min (X,Y) % % X%Y : 100*[(X/Y) 1] > X > Y : 1 X Y, 0

>= X >= Y : 1 X Y, 0 < X < Y : 1 X Y, 0

LIMDEP : 25

LIMDEP LIMDEP Help LIMDEP

Log(x) Natural logarithm Exp(x) Exponential Abs(x) Absolute Value Sqr(x) Square root Sin(x) Sine Rsn(x) Arcsine Cos(x) Consine Rcs(x) Arccosine Tan(x) Tangent Gma(x) Gamma Function Phi(x) CDF of standard nomal N01(x) PDF of standard nomal Lmm(x) -N01/Phi Lmp(x) N01/(1-Phi) Inp(x) Inverse normal CDF Inf(x) Inverse normal PDF Lgt(x) Logit [log(z/(1-z))] Lgp(x) Logistic CDF [exp(x)/(1+exp(x))] Lgd(x) Logistic density [Lgp(1-Lgp)] Trn(x1,x2) Trend Ind(i1,i2) Dummy Variable [ 1 i1 observation number i2, 0] Dmy(p,i1) Seasonal dummy [ 1 pth i1 0] d(x) x [xt xt-1] x[-1] Lag variable [Xt-1]

LIMDEP

Help LIMDEP

LIMDEP : 26

LIMDEP CREATE 1 Functions Log(x) 2 ^ @ 3 *, /, !, ~, %, >, >=,

LIMDEP : 27

3 LIMDEP HISTOGRAM Opitons HISTOGRAM [ Options ]

4 LIMDEP Histogram Plot

LIMDEP : 28

Output

Command Bar Command Document

HISTOGRAM ; Rhs= [] $ HISTOGRAM ; Rhs=Y$

HISTOGRAM ; Rhs = [1, 2,.....]

Plot Variable

Opitions , Linear regression, Grid

1 Project Demo.lpj 2 Model/Data Description/Plot Variables Main Mnus

LIMDEP : 29

3 LIMDEP PLOT

LIMDEP : 30

4 Options Options Opitons PLOT [ Options ] Opitions [ Linear regression Grid]

Options

5 LIMDEP Plot

LIMDEP : 31


PLOT ; Lhs = [ X] ; Rhs = [ Y ( 5 )] ; Option $

Options Title ......... ; Title = Titile ; Fill Linear regression ; Regression Grid ; Grid Fixed X ; Endpoints = , Fixed X ; Limits = ,

PLOT ; Lhs = X1 ; Rhs = Y ; Regression; Grid $

Multiple Scatter Plots Scatter Plots

1 Project Demo.lpj 2 Model/Data Description/Multiple Scatter Plot Main Menus

LIMDEP : 32

3 LIMDEP SPLOT

4 LIMDEP Scatter Plots

LIMDEP : 33


SPLOT ; Rhs = [ 1, 2,] $

SPLOT ; Rhs = Y, X1, X2, X3 $

File/Save File/Save As Main Menus Ctrl S Plot [] Tool Bar LIMDEP File Options 2 Windows Metafile (*.wmf) Windows Bitmap (*.bmp)

3.5

LIMDEP Command Wizard

1 Project Demo.lpj 2 Model/Data Description/Descriptive Statistics Main Menus

LIMDEP : 34

3 LIMDEP DSTATS

4 Options LIMDEP 4 (Mean), (Std.Dev.), (Minimum) (Maximum) Output

4 LIMDEP Options

Options Opitons DSTATS

LIMDEP : 35

[ LIMDEP Options ] Opitions

[ Correlation matrix] Options

5 LIMDEP Output


DSTAT ; Rhs = [ 1, 2,... ] ; Option $

LIMDEP : 36

Options Covariance matrix ; Output = 1 Correlation matrix ; Output = 2 Covariance Correlation matrix ; Output = 3 Skewness and kurtosis ; All First Oreder Autocorrelation ; AR1 Bowman and Shenton chi-squared ; Normality test ......... ; Wts = weight ..... ; Str =

DSTAT ; Rhs = Y, X1, X2, X3 ; Output = 2 $

4. (Ordinary Least Square Method)

[ OLS] [: Error term ( e /)] OLS [ OLS]

4.1

. [ iii xy ++= ]

. ( Non Stochastic Variable) .

[ )1X,X(Corr ji ] [ (Correlation Coefficients) 0.8] Multicollinearity

. (Error term) [ ),0(N~ 2i , )(E i = 0 22i )(E = Homoskedasticity]

. [ 0 = ),( E = ),(Cov jiji i j] Autocorrelation

.

LIMDEP : 37

4.2

(Linear)

Linear 2211 XXY ++= Double Log 2211 XlnXlnYln ++= Linear Log 2211 XXlnY ++= Log Linear 2211 XXYln ++= Polynomial 23

21211 XXXY +++=

Inverse 2211 X)X/1(Y ++= Dummy 1211 DXY ++= Dummy (Interaction with Variable) 1131211 XDDXY +++=

: R.R. Johnson (2000). 2R , 2R (Adjusted 2R ), Fstatistic CIA Akaikes Information Criterion

2R ( ) ( )yyyy

1R 2

=

2R (Adjusted 2R ) ( ) kT1TR11R 22

= 2R 2R

F statistic

( )( )

=1kn/

k/yyF 22

F-statistic P-value F-statistic <

CIA (Akaikes Information Criterion)

T/k2lT2AIC +=

LIMDEP : 38

4.3 LIMDEP

LIMDEP (Y) (SE), (F) (L) 1,000

++++= LNLFlnSElnYln LFSE Yln = Natural logarithm SEln = Natural logarithm Fln = Natural logarithm Lln = Natural logarithm

LFSE ,,, = = LIMDEP OLS 1 LIMDEP 3.1 18 20 Data Editor

2 1 Natural logarithm Transform

3.3 23 26 CREATE; LNY=log(Y);LNSE=log(SE);lNF=LOG(F); lnL=log(L)$ Command Bar Command Document Run LIMDEP Transform

LIMDEP : 39

3 Model/Linear Models/Regression Main Menus

LIMDEP : 40

4 LIMDEP REGRSS

5

Dependent variable Menu button [ ] LNY

Independent variables c 1 d LNSE, LNF LNL ONE

6 LIMDEP Output

Command Bar Command Document LIMDEP OLS

REGRESS ; Lhs = [] ; Rhs = [] ; Option [] $

REGRESS ; Lhs=LNY ; Rhs = ONE, LNSE, LNF, LNL$

REGRESS ; Lhs=LOG(Y) ; Rhs = ONE, LOG(SE), LOG(F), LOG(L)$

1 2

LIMDEP : 41

LIMDEP OLS

y = Xb + e

y Vector X Matrixs e Vector b Vector

OLS (b) (e)

b = (XX)-1 Xy e = y - Xb

LIMDEP +----------------------------------------------------+

| Ordinary least squares regression | | Model was estimated Oct 11, 2007 at 03:45:44PM | | LHS=LNY Mean = 6.762841 | | Standard deviation = .1401046 | | WTS=none Number of observs. = 1000 | | Model size Parameters = 4 | | Degrees of freedom = 996 | | Residuals Sum of squares = 13.94081 | | Standard error of e = .1183081 | | Fit R-squared = .2890848 | | Adjusted R-squared = .2869435 | | Model test F[ 3, 996] (prob) = 135.00 (.0000) | | Diagnostic Log likelihood = 717.5288 | | Restricted(b=0) = 546.9277 | | Chi-sq [ 3] (prob) = 341.20 (.0000) | | Info criter. LogAmemiya Prd. Crt. = -4.264935 | | Akaike Info. Criter. = -4.264935 | | Autocorrel Durbin-Watson Stat. = 2.0255879 | | Rho = cor[e,e(-1)] = -.0127940 | +----------------------------------------------------+

+---------+--------------+----------------+--------+---------+----------+

|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| +---------+--------------+----------------+--------+---------+----------+

Constant 5.55510058 .06864247 80.928 .0000 LNSE .20272842 .01892009 10.715 .0000 3.09323414 LNF .10040294 .00858633 11.693 .0000 3.31983699 LNL .13073853 .01052616 12.420 .0000 1.89181432

LIMDEP 2

1:

[Ordinary least squares regression] [Model was estimated Oct 11, 2007 at 03:45:44PM]

LIMDEP : 42

[LHS = ..] o Mean [

== n

1iiy)n/1(y ]

o Standard deviation [ =

n1i

2/12i })yy()]{1n/(1{[ ]

[WTS = ..] o [Number of observes = n]

(Model size) o [Parameters = K] o Degrees of freedom [n K]

(Residuals) o Sum of squares [

==== n

1i

2ii

n

1i

2ii )b'xy()yy(e'e ]

o Standard error of e [ )Kn/(e'es = ] Goodness of fit

o RSquared [ =

= n1i

2ii

2 )b'xy(/e'e1R ]

o Adjusted Rsquared [ ]R1[0Kn/()1n(1R 22 = ] (Model test)

o F-statistic [ )]kn/()R1/[()]1K/(R[]Kn,1K[F 22 = ] o Prob value of F [ Fobserved)]kn,1K(F[obProbPr F >= ]

Diagnostic o Log Likelihood [ )]n/e'elog(2log1[2/nLlog ++= ] o Restrited Log Likelihood [ )]slog(2log1[2/nLlog 2y0 ++= ]

[ =

= n1i

2i

2y )yy()n/1(s ]

o Chi-square [K-1] [ )LogLLogL(2 02 = ] o Prob value of 2 [ 22 observed)]1K([obProbPr 2 >= ]

Information criterion o Log Amemiya Prediction Criterion [ )]n/K1(slog[APC 2 += ] o Akaike Information Criterion [ )2log1()2/n/()kL(logAIC += ]

Autocorrelation o Dubin-Watson Statistic [

===

T

2t

2t

T

2t

21tt e/)ee(DW ]

o Rho [ 2/dw1r = ]

LIMDEP : 43

2:

6 1 (Variable) 2 (Coefficient) [ Y'X)X'X(b 1= ] 3 (Standard Error) [ 12 )X'X(sse = ] 4 t (t-statistic) 2 3 [ kkk se/bt = ] 5 Probability value (P[| T | > t]) 0 1 t-statistic

t-statistic P-value () P-value < [P-value = Prob[t(n-K)] > observed tk]

6 (Mean of X)

4.4 LIMDEP Stepwise

OLS LIMDEP Options Stepwise

1 1 5 4.3 38 40 Options

LIMDEP : 44

2 Stepwise regression LIMDEP Output

Command Bar Command Document Options Stepwise regression OLS

REGRESS ; Lhs = [] ; Rhs = [] ; Option [] ; Alg = Step $

REGRESS ; Lhs = LNY ; Rhs = ONE, LNSE, LNF, LNL ; Alg = Step $

LIMDEP Stepwise Step LIMDEP Mallows Cp Mallows Cp

T)1P(2s/e'eCp 2 ++= LIMDEP

+-------------------------------------------------------------------------+

| Stepwise Regression | | Dependent variable = LNY | | Number of observations = 1000 | | Number of regressors = 3 | | Degrees of freedom = 996 | | Predictor variables are: | | LNSE LNF LNL | +-------------------------------------------------------------------------+

+----------------------------------------------------+

| Step 1 of Stepwise Regression | | Ordinary least squares regression | | Model was estimated Oct 15, 2007 at 11:33:49AM | | LHS=LNY Mean = 6.762841 | | Standard deviation = .1401046 | | WTS=none Number of observs. = 1000 | | Model size Parameters = 2 | | Degrees of freedom = 998 | | Residuals Sum of squares = 17.48302 | | Standard error of e = .1323558 | | Fit R-squared = .1084487 | | Adjusted R-squared = .1075554 | | Model test F[ 1, 998] (prob) = 121.40 (.0000) | | Diagnostic Log likelihood = 604.3239 | | Restricted(b=0) = 546.9277 | | Chi-sq [ 1] (prob) = 114.79 (.0000) | | Info criter. LogAmemiya Prd. Crt. = -4.042525 | | Akaike Info. Criter. = -4.042525 | | Mallows Cp = 257.073 | +----------------------------------------------------+

LIMDEP : 45

+------------------------------------------------------------------+

| Analysis of Variance for the Current Regression | | Source Deg.Fr. Sum of squares Mean Square F | | Regression 1 2.12664 2.12664 121.40 | | Residual 998 17.48302 .01752 | | Total 999 19.60967 .01963 | | Variable entered this step = LNF , Deleted = | +------------------------------------------------------------------+

+---------+--------------+----------------+--------+---------+----------+


LNF .10573405 .00959645 11.018 .0000 3.31983699 Constant 6.41182163 .03213240 199.544 .0000

+----------------------------------------------------+


+------------------------------------------------------------------+

| Analysis of Variance for the Current Regression | | Source Deg.Fr. Sum of squares Mean Square F | | Regression 2 4.06187 2.03094 130.23 | | Residual 997 15.54780 .01559 | | Total 999 19.60967 .01963 | | Variable entered this step = LNL , Deleted = | +------------------------------------------------------------------+

+---------+--------------+----------------+--------+---------+----------+

|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| +---------+--------------+----------------+--------+---------+----------

LNF .10140136 .00906264 11.189 .0000 3.31983699 LNL .12351794 .01108793 11.140 .0000 1.89181432 Constant 6.19253246 .02917511 212.254 .0000

LIMDEP : 46

+----------------------------------------------------+


+------------------------------------------------------------------+

| Analysis of Variance for the Current Regression | | Source Deg.Fr. Sum of squares Mean Square F | | Regression 3 5.66886 1.88962 135.00 | | Residual 996 13.94081 .01400 | | Total 999 19.60967 .01963 | | Variable entered this step = LNSE , Deleted = | | **********> This is the final equation z] | Mean of X| +---------+--------------+----------------+--------+---------+----------+

LNF .10040294 .00858633 11.693 .0000 3.31983699 LNL .13073853 .01052616 12.420 .0000 1.89181432 LNSE .20272842 .01892009 10.715 .0000 3.09323414 Constant 5.55510058 .04592166 120.969 .0000

Stepwise LIMDEP

OLS OLS LIMDEP

Lln1307.0Fln1004.0SEln2027.05551.5Yln +++= t-statistic (80.928) (10.715) (11.693) (12.420)

R2 = 0.2891 2R = 0.2869 F-statistic [3, 996] = 135 (Prob. = 0.0000)

28.91% [ R2] 99% [ F-statistic 99% Prob.

LIMDEP : 47

4.5

/ (Prdictions) (Residuals) LIMDEP

1 1 5 4.3 38 40 Output

2 Display [LIMDEP Predictions Residuals Output]

Keep [LIMDEP Predictions Residuals Data Editor] Keep Predictoins as variable Ypre Keep Residuals as variable Resid

LIMDEP Data Editor

LIMDEP : 48

Command Bar Command Document Options 2 OLS

REGRESS ; Lhs = [] ; Rhs = [] ; Keep = [] ; Res = [] $

REGRESS ; Lhs = LNY ; Rhs=ONE, LNSE, LNF, LNL ; Keep = Ypre ; Res = Resid $

LIMDEP Plot residuals REGRESS Output

Command Bar Command Document Options ; Plot

OLS

REGRESS ; Lhs = [] ; Rhs = [] ; Plot$

REGRESS ; Lhs = LNY ; Rhs = ONE, LNSE, LNF, LNL ; Plot $

Enter LIMDEP

LIMDEP : 49

LIMDEP

Standardize rediduals befor plotting or keeping Plot rediduals REGRESS Output Command Bar Command Document Options ; Standardize

; Plot OLS

REGRESS ; Lhs = [] ; Rhs = [] ; Standardize ; Plot $

REGRESS ; Lhs = LNY ; Rhs = ONE, LNSE, LNF, LNL ; Standardize ; Plot $

LIMDEP : 50

Enter LIMDEP

Standardize rediduals LIMDEP Belsley, Kuh

and Welsh (1980)

2/1ii2iiii

2/1ii

2ii )]1Kn/()h1/(ee'e/[()]h1/(e[)]h1(s/[eu ==

i1

iii x)X'X('xh=

4.6

LIMDEP LIMDEP Equality Restrictions Inequality Restrictions Equality Restrictions LIMDEP

LIMDEP : 51

Linear regression

eXy += Subject to qR R Matrix JK J Linearly independent restriction

X'X Nonsingular

]qRb[]'R)X'X(R['R]X'X[bb 111c = b y'X]X'X[ 1=

1111212c ]X'X[R]'R]X'X[R['R]X'X[s]X'X[s]b[Var =

2s )JKn/()Xby()'Xby( cc = J restrictions F-statistic

R:H0 = q F-statistic ]Kn,J[F = J/]qRb[]'R)X'X(Rs[]'qRb[ 112 = )]Kn(e~'e~/[]J/)e'ee~'e~[( e~ = b =

LIMDEP

++++= LlnFlnSElnYln LFSE

[ ] 1 [ 1LFSE =++ ] LIMDEP

LIMDEP : 52

1 1 5 4.3 38 40 Options

2 Impose and test the restrictions b(2)+b(3)+b(4)=1 LIMDEP Output

Command Bar Command Document Options OLS

REGRESS ; Lhs = [] ; Rhs = [] ; Cls : [] $

REGRESS ; Lhs = LNY ; Rhs = ONE, LNSE, LNF, LNL ; Cls : b(2)+b(3)+b(4) = 0 $

Enter LIMDEP (F-statistic)

LIMDEP : 54

[ ] 1 99% [ Restrictions F-statistic 99% Prob.

LIMDEP : 55

1 1 3 3.5 33 34 Data Desciption /Descrptive Statistics Main Menus [ (one)] Options

2 Display correlation matrix LIMDEP Output

Command Bar Command Document Option ; Output = 2 OLS

DSTAT ; Rhs = [ 1, 2,... ] ; Output = 2 $

DSTAT ; Rhs = LNY, LNSE, LNF, LNL ; Output = 2 $

Enter LIMDEP Correlation matrix

Descriptive Statistics All results based on nonmissing observations. ===============================================================================

Variable Mean Std.Dev. Minimum Maximum Cases ===============================================================================

-------------------------------------------------------------------------------

All observations in current sample -------------------------------------------------------------------------------

LNY 6.76284145 .140104595 6.29194685 7.15329417 1000 LNSE 3.09323414 .198250537 2.70829684 3.40040373 1000 LNF 3.31983699 .436364830 2.30399410 3.91168695 1000 LNL 1.89181432 .356659461 1.10135851 2.39629399 1000

LIMDEP : 56

Correlation Matrix for Listed Variables

LNY LNSE LNF LNL LNY 1.00000 .26822 .32932 .32799 LNSE .26822 1.00000 .00809 -.06361 LNF .32932 .00809 1.00000 .04292 LNL .32799 -.06361 .04292 1.00000

Variance Inflation Factors (VIF) VIF Multicollinearity VIF

5 Multicollinearity (Studenmund 2006: 259) 10 ( , 2546) VIF

( )2kk R11VIF =

kk2kn

1iikk )X'X()xx(VIF

=

=

k

Command Document LIMDEP VIF

NAMELIST ; x = [] $

MATRIX ; List ; xm0x = {n-1}*Xvcm(x) ; vif = Diag () * vecd(xm0x)$

NAMELIST ; x = ONE, LNSE, LNF, LNL $

MATRIX ; List ; xm0x = {n-1}*Xvcm(x) ; vif = Diag () * vecd(xm0x)$

Hightlight LIMDEP VIF Output

Matrix VIF has 4 rows and 1 columns. 1 +-------------- 1| .0000000D+00 2| 1.00418 3| 1.00196 4| 1.00597

LNSE LNF LNL

VIF = 1.00418 VIF = 1.00196 VIF = 1.00597

LIMDEP : 57

Multicollinearity Simple Correlation Coefficients Variance Inflation Factors (VIF) Multicollinearity 3 0.80 VIF 3 5

5.3 Multicollinearity

. Multicollinearity Multicollinearity Bias t-statistic

. Multicollinearity

.

. (Transforming) Multicollinearity EViews

Linear combination Yt = 0 + 1(Xt + Zt) First difference Yt = 0 + 1(Xt - Xt-1) First difference of the logarithm Yt = 0 + 1(lnXt - lnXt-1) One-period % change (in decimal) Yt = 0 + 1[(Xt - Xt-1)/ Xt]

: R.R. Johnson (2000).

6. Heteroskedasticity

6.1 Heteroskedasticity

(Error /Residuals: ) [ 22i )(E ] [OLS] [ 22i )(E = ] 2 . (Impure heteroskedasticity) . (Pure heteroskedasticity)

LIMDEP : 58

(Cross sectional data) [Time series data]

Heteroskedasticity Unbiased Consistency Efficiency OLS Heteroskedasticity t-statistic


LIMDEP Heteroskedasticity Brusch and Pagen Lagrange multiplier test

1 1 5 4.3 38 40 Options

2 Robust VC matrix/Phil-Perron test hetero. (White)

LIMDEP Output

Command Bar Command Document Option ; Het OLS

LIMDEP : 59

REGRESS ; Lhs = [] ; Rhs = [] ; Het $

REGRESS ; Lhs = LNY ; Rhs = ONE, LNSE, LNF, LNL ; Het $

LIMDEP +----------------------------------------------------+

| Ordinary least squares regression | | Model was estimated Oct 16, 2007 at 02:37:41PM | | LHS=LNY Mean = 6.762841 | | Standard deviation = .1401046 | | WTS=none Number of observs. = 1000 | | Model size Parameters = 4 | | Degrees of freedom = 996 | | Residuals Sum of squares = 13.94081 | | Standard error of e = .1183081 | | Fit R-squared = .2890848 | | Adjusted R-squared = .2869435 | | Model test F[ 3, 996] (prob) = 135.00 (.0000) | | Autocorrel Durbin-Watson Stat. = 2.0255879 | | Rho = cor[e,e(-1)] = -.0127940 | | White heteroscedasticity robust covariance matrix | | Br./Pagan LM Chi-sq [ 3] (prob) = 24.58 (.0000) | +----------------------------------------------------+

+---------+--------------+----------------+--------+---------+----------+



H0: Homoscedasticity

H1: Heteroskedasticity

Chi-square 99% [Prob. < ] Heteroskedasticity


Heteroskedasticity Heteroskedasticity 2 Heteroskedasticity-Corrected Standard Errors Weighted Least Square (WLS)

Heteroskedasticity-Corrected Standard Errors LIMDEP Options 4 . Whites (1978)

Options ; Het 1n

1iii

2i

1 )X'X('xxe)X'X(]b[Var =

=

LIMDEP : 60

. Davidson and Mackinno (1993) White 3

Options ; Het ; Hc1 1n

1iii

2i

1 )X'X('xxekn

n)X'X(]b[Var =

= Options ; Het ; Hc2 1

n

1iii

i1

i

2i1 )X'X('xx

)x)X'X(x1(e)X'X(]b[Var

= =

Options ; Het ; Hc3 1n

1iii2

i1

i

2i1 )X'X('xx

)x)X'X(x1(e)X'X(]b[Var

= =

LIMDEP 4 1 1 5 4.3 38 40 Options

2 Robust VC matrix/Phil-Perron test

LIMDEP Output Command Bar Command Document

Option Heteroskedasticity 4 OLS

REGRESS ; Lhs = [] ; Rhs = [] ; Het ; [Hc1, Hc2, Hc1] $

LIMDEP : 61

REGRESS ; Lhs=LNY ; Rhs = ONE, LNSE, LNF, LNL ; Het $

REGRESS ; Lhs=LNY ; Rhs = ONE, LNSE, LNF, LNL ; Het ; Hc1 $



LIMDEP +----------------------------------------------------+

| Ordinary least squares regression | | Model was estimated Oct 16, 2007 at 03:33:25PM | | LHS=LNY Mean = 6.762841 | | Standard deviation = .1401046 | | WTS=none Number of observs. = 1000 | | Model size Parameters = 4 | | Degrees of freedom = 996 | | Residuals Sum of squares = 13.94081 | | Standard error of e = .1183081 | | Fit R-squared = .2890848 | | Adjusted R-squared = .2869435 | | Model test F[ 3, 996] (prob) = 135.00 (.0000) | | Autocorrel Durbin-Watson Stat. = 2.0255879 | | Rho = cor[e,e(-1)] = -.0127940 | | White heteroscedasticity robust covariance matrix | | Br./Pagan LM Chi-sq [ 3] (prob) = 24.58 (.0000) | +----------------------------------------------------+

REGRESS ; Lhs=LNY ; Rhs = ONE, LNSE, LNF, LNL $ +---------+--------------+----------------+--------+---------+----------+



REGRESS ; Lhs=LNY ; Rhs = ONE, LNSE, LNF, LNL ; Het $ +---------+--------------+----------------+--------+---------+----------+



REGRESS ; Lhs=LNY ; Rhs = ONE, LNSE, LNF, LNL; Het ; Hc1 $ +---------+--------------+----------------+--------+---------+----------+



REGRESS ; Lhs=LNY ; Rhs = ONE, LNSE, LNF, LNL; Het ; Hc2 $ ---------+--------------+----------------+--------+---------+----------+



LIMDEP : 62

REGRESS ; Lhs=LNY ; Rhs = ONE, LNSE, LNF, LNL; Het ; Hc3 $ +---------+--------------+----------------+--------+---------+----------+



Heteroskedasticity [ Standard Error] t-statistic [ t-statistic Coefficient Standard Error] Heteroskedasticity Heteroskedasticity

Weighted Least Square (WLS) Z Weight

LIMDEP 1 CREATE ; wts = 1/z^2$ Command Bar 2 1 5 4.3 38 40 Main

2 Weight using variable WTS []

LIMDEP Output

LIMDEP : 63

Command Bar Command Document Option OLS

REGRESS ; Lhs = [] ; Rhs = [] ; Wts = [] $

REGRESS ; Lhs=LNY ; Rhs = ONE, LNSE, LNF, LNL ; Wts = wts $

LIMDEP +----------------------------------------------------+

| Ordinary least squares regression | | Model was estimated Oct 16, 2007 at 04:11:51PM | | LHS=LNY Mean = 6.742803 | | Standard deviation = .1458603 | | WTS=WTS Number of observs. = 1000 | | Model size Parameters = 4 | | Degrees of freedom = 996 | | Residuals Sum of squares = 14.97227 | | Standard error of e = .1226067 | | Fit R-squared = .2955537 | | Adjusted R-squared = .2934319 | | Model test F[ 3, 996] (prob) = 139.29 (.0000) | | Diagnostic Log likelihood = 636.8849 | | Restricted(b=0) = 461.7133 | | Chi-sq [ 3] (prob) = 350.34 (.0000) | | Info criter. LogAmemiya Prd. Crt. = -4.193555 | | Akaike Info. Criter. = -4.103647 | | Autocorrel Durbin-Watson Stat. = 2.0101984 | | Rho = cor[e,e(-1)] = -.0050992 | +----------------------------------------------------+

+---------+--------------+----------------+--------+---------+----------+



WTS WTS WTS

7. Autocorrelation

7.1 Autocorrelation

(Error /Residuals: ) Heteroskedasticity Autocorrelation [ 0),(E),(Cov jiji = i j] (OLS) [ 0),(E),(Cov jiji == i j]

LIMDEP : 64

[ Positive Autocorrelation] [ Negative Autocorrelation] [ Serial Correlation] [ Spatial Correlation] 2 . Impure Autocorrelation (Specification Error) , Cobweb, Lagged variables . Pure Autocorrelation

Autocorrelation Unbiased Efficiency () OLS BLUE Autocorrelation Autocorrelation (Underestimate) t-statistic

7.2 Autocorrelation

Autocorrelation Durbin-Watson (D.W.), Durbin-Watson (D.W.) LIMDEP 2 [ 41 43] 1 LIMDEP Durbin-Watson statistic +----------------------------------------------------+

| Ordinary least squares regression | | Model was estimated Oct 21, 2007 at 04:44:38PM | | LHS=LNY Mean = 6.762841 | | Standard deviation = .1401046 | | WTS=none Number of observs. = 1000 | | Model size Parameters = 4 | | Degrees of freedom = 996 | | Residuals Sum of squares = 13.94081 | | Standard error of e = .1183081 | | Fit R-squared = .2890848 | | Adjusted R-squared = .2869435 | | Model test F[ 3, 996] (prob) = 135.00 (.0000) | | Diagnostic Log likelihood = 717.5288 | | Restricted(b=0) = 546.9277 | | Chi-sq [ 3] (prob) = 341.20 (.0000) | | Info criter. LogAmemiya Prd. Crt. = -4.264935 | | Akaike Info. Criter. = -4.264935 | | Autocorrel Durbin-Watson Stat. = 2.0255879 | | Rho = cor[e,e(-1)] = -.0127940 | +----------------------------------------------------+

+---------+--------------+----------------+--------+---------+----------+



LIMDEP : 65

Durbin-Watson Test (First Order Autocorrelation)

H0: = 0 (Non Autocorrelation) H1: 0 (Autocorrelation)

Durbin Watson statistic

( )

=

=

= T1t

2t

T

2t

21tt

.W.D

OLS

( )= 12.W.D

[ t1tt u+= ]

=

=

= T1t

2t

T

2t1tt

D.W. (Durbin Watson statistic) 0 4 [0 Positive Autocorrelation 4 Negative Autocorrelation]

= -1 D.W. = 4 Perfect Negative Autocorrelation = 0 D.W. = 2 Autocorrelation = 1 D.W. = 0 Perfect Positive Autocorrelation D.W. 2

Autocorrelation D.W. Durbin-Watson Test Durbin Watson Statistic

dL > D.W. > 4 - dL (H0) Autocorrelation 4 dU > D.W. > dU (H0) Autocorrelation

LIMDEP : 66

Autocorrelation

: (2546).

D.W. 2.026 Durbin-Watson n=1,000 [n n = 1,000 Durbin-Watson n = 1,000 n=200 ] k = 3 [k k = 3 ] dL = 1.643 dU = 1.704 D.W. dU [1.704] < 2.026 H0: Positive Autocorrelation Autocorrelation

: Autocorrelation Time series data Cross section data Autocorrelation

7.3 Autocorrelation

Autocorrelation LIMDEP Autocorrelation Autocorrelation Prais-Winsten algorithm, Cochrane-Orcult algorithm, Grid search MLE (Beach and MacKinnon, 1978) Autocorrelation Prais-Winsten algorithm Cochrane-Orcult algorithm First order autocorrelation

ttt x'y += t1tt u+= Autocorrelation

Autocorrelation Prais-Winsten algorithm Cochrane-Orcult algorithm

t1tt*

1tt )xx(yy += Prais-Winsten algorithm Cochrane-Orcult algorithm 0.0001 LIMDEP Autocorrelation Autocorrelation LIMDEP Autocorrelation Autocorrelation Prais-Winsten algorithm

0 dL dU 4-dL 2 4-dU 4 H0 H0 H0

LIMDEP : 67

Cochrane-Orcult algorithm OLS 4.3 39 44 +----------------------------------------------------+

| Ordinary least squares regression | | Model was estimated Oct 21, 2007 at 09:37:29PM | | LHS=G Mean = .7935996 | | Standard deviation = .1702296 | | WTS=none Number of observs. = 52 | | Model size Parameters = 6 | | Degrees of freedom = 46 | | Residuals Sum of squares = .1998129E-01 | | Standard error of e = .2084169E-01 | | Fit R-squared = .9864798 | | Adjusted R-squared = .9850102 | | Model test F[ 5, 46] (prob) = 671.26 (.0000) | | Diagnostic Log likelihood = 130.6845 | | Restricted(b=0) = 18.79165 | | Chi-sq [ 5] (prob) = 223.79 (.0000) | | Info criter. LogAmemiya Prd. Crt. = -7.632401 | | Akaike Info. Criter. = -7.633433 | | Autocorrel Durbin-Watson Stat. = .4547690 | | Rho = cor[e,e(-1)] = .7726155 | +----------------------------------------------------+

+---------+--------------+----------------+--------+---------+----------+

|Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] | Mean of X| +---------+--------------+----------------+--------+---------+----------+

Constant -6.60345553 .25184573 -26.220 .0000 PG -.08858575 .01725380 -5.134 .0000 3.72930296 Y .80406937 .02982440 26.960 .0000 9.67487347 PNC .00095098 .00087632 1.085 .2835 87.5673077 PUC -.00067018 .00056622 -1.184 .2426 77.8000000 PPT -.00092835 .00024897 -3.729 .0005 89.3903846

D.W. 0.455 D.W. 0.455 < dL = 1.623 H0: Positive Autocorrelation Autocorrelation Autocorrelation

1 1 5 4.3 38 40 Options

LIMDEP : 68

2 Autocorrelation 2 Correct for autocorrelation using

LIMDEP Output Command Bar Command Document

Options Aotocorrelation OLS

Prais-Winsten algorithm

REGRESS ; Lhs = [] ; Rhs = [] ; AR1 $

Cochrane-Orcult algorithm

REGRESS ; Lhs = [] ; Rhs = [] ; AR1 ; Alg = C$

REGRESS ; Lhs=G ; Rhs = ONE, PG, Y, PNC, PUC, PPT ; AR1 $

REGRESS ; Lhs=G ; Rhs = ONE, PG, Y, PNC, PUC, PPT ; AR1 ; Alg = C$

LIMDEP

LIMDEP : 69

Autocorrelation +----------------------------------------------------+

| Ordinary least squares regression | | Model was estimated Oct 21, 2007 at 09:58:02PM | | LHS=G Mean = .7935996 | | Standard deviation = .1702296 | | WTS=none Number of observs. = 52 | | Model size Parameters = 6 | | Degrees of freedom = 46 | | Residuals Sum of squares = .1998129E-01 | | Standard error of e = .2084169E-01 | | Fit R-squared = .9864798 | | Adjusted R-squared = .9850102 | | Model test F[ 5, 46] (prob) = 671.26 (.0000) | | Diagnostic Log likelihood = 130.6845 | | Restricted(b=0) = 18.79165 | | Chi-sq [ 5] (prob) = 223.79 (.0000) | | Info criter. LogAmemiya Prd. Crt. = -7.632401 | | Akaike Info. Criter. = -7.633433 | | Autocorrel Durbin-Watson Stat. = .4547690 | | Rho = cor[e,e(-1)] = .7726155 | +----------------------------------------------------+

+---------+--------------+----------------+--------+---------+----------+

|Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] | Mean of X| +---------+--------------+----------------+--------+---------+----------+

Constant -6.60345553 .25184573 -26.220 .0000 PG -.08858575 .01725380 -5.134 .0000 3.72930296 Y .80406937 .02982440 26.960 .0000 9.67487347 PNC .00095098 .00087632 1.085 .2835 87.5673077 PUC -.00067018 .00056622 -1.184 .2426 77.8000000 PPT -.00092835 .00024897 -3.729 .0005 89.3903846

Prais-Winsten algorithm +---------------------------------------------+

| AR(1) Model: e(t) = rho * e(t-1) + u(t) | | Initial value of rho = .77262 | | Maximum iterations = 100 | | Method = Prais - Winsten | | Iter= 1, SS= .008, Log-L= 155.624004 | | Iter= 2, SS= .007, Log-L= 156.148862 | | Iter= 3, SS= .007, Log-L= 156.318414 | | Iter= 4, SS= .007, Log-L= 156.392576 | | Iter= 5, SS= .007, Log-L= 156.424530 | | Iter= 6, SS= .007, Log-L= 156.436523 | | Final value of Rho = .918975 | | Iter= 10, SS= .007, Log-L= 156.437396 | | Durbin-Watson: e(t) = .162051 | | Std. Deviation: e(t) = .031643 | | Std. Deviation: u(t) = .012477 | | Durbin-Watson: u(t) = 1.658846 | | Autocorrelation: u(t) = .170577 | | N[0,1] used for significance levels | +---------------------------------------------+

+---------+--------------+----------------+--------+---------+----------+


Constant -5.44283063 .53376460 -10.197 .0000 PG -.10034734 .01911771 -5.249 .0000 3.72930296 Y .68335744 .05841719 11.698 .0000 9.67487347 PNC .00015758 .00094358 .167 .8674 87.5673077 PUC .923510D-04 .00043412 .213 .8315 77.8000000 PPT -.00036736 .00044573 -.824 .4098 89.3903846 RHO .91897461 .05521538 16.643 .0000

LIMDEP : 70

LIMDEP Autocorrelation Prais-Winsten algorithm D.W. 1.659 [ Rho Tranform = 0.9819] D.W. 1.659 > dL = 1.623 Autocorrelation Cochrane-Orcult algorithm +---------------------------------------------+

| AR(1) Model: e(t) = rho * e(t-1) + u(t) | | Initial value of rho = .77262 | | Maximum iterations = 100 | | Method = Cochrane - Orcutt | | Iter= 1, SS= .007, Log-L= 157.462781 | | Iter= 2, SS= .006, Log-L= 159.330987 | | Iter= 3, SS= .006, Log-L= 159.379899 | | Iter= 4, SS= .007, Log-L= 156.333441 | | Final value of Rho = .965332 | | Iter= 4, SS= .007, Log-L= 156.333441 | | Durbin-Watson: e(t) = .000021 | | Std. Deviation: e(t) = .045305 | | Std. Deviation: u(t) = .011826 | | Durbin-Watson: u(t) = 1.330149 | | Autocorrelation: u(t) = .334925 | | N[0,1] used for significance levels | +---------------------------------------------+

+---------+--------------+----------------+--------+---------+----------+


Constant -3.28269625 .94719591 -3.466 .0005 PG -.11126575 .01832563 -6.072 .0000 3.72930296 Y .48557042 .09112087 5.329 .0000 9.67487347 PNC -.00058725 .00095036 -.618 .5366 87.5673077 PUC .00022585 .00040891 .552 .5807 77.8000000 PPT -.00018424 .00044149 -.417 .6765 89.3903846 RHO .96533247 .03655059 26.411 .0000

Autocorelation Cochrane-Orcutt algorithm D.W. 1.330 [ Rho Tranform = 0.9653] D.W. 1.330 < dL = 1.623 Autocorelation Cochrane-Orcutt algorithm Autocorrelation Autocorrelation Prais-Winsten algorithm

8. Binary Choice Models

, 2 Binary choice 0 1 [ 0 1] 0 1 3 (Linear Probability Model), (Probit Model)

LIMDEP : 71

(Logit Model) (Linear Probability Model)

. (Nonnormality of Distribution) . (Heteroskedasticity) . Y [ Y ] 0 1 (0 E(Y|X) 1) . R2

OLS (Probit and Logit Model) Binary MLE (Maximum Likelihood Estimation)

. Binary Response Dummy Variable /Interval /Ratio scale . () 0 [E(i) = 0] . [Cov(ij) = 0] . . . 30*P [n 30*P] [P Parameter] ( ) ( )== ii 'xFX|1yprob Probit Probit Model ( ) dz

2zexp

21x'x1yprob

2i

i

i

=

==

( )

==

ii

'x1yprob

Logistic Logit Model ( ) xx

+== e1e1yprob i

Probit Logit Probit (Normal Distribution) Logit (Logistic Distribution)

LIMDEP : 72

Probability Density Function Probit Logit Cumulative Distribution function Probit Logit Probit Logit

: 766 [ Probit /Logit Model 30 1 ]

+++++++= 443322112211 DDDDXXY Y = Y = 1 Y = 0 X1 = () X2 = (/) D1 = D1 = 1 D1 = 0 D2 = D2 = 1 D2 = 0 D3 = D3 = 1 D3 = 0

D4 = D4 = 1 D4 = 0 =

Density Function

Z

.087

.170

.253

.336

.419

.004-1.80 -.60 .60 1.80 3.00-3.00

PROBIT LOGIT

PD

F

Probit

Logit

Probability Function

Z

.21

.42

.63

.84

1.05

.00-1.80 -.60 .60 1.80 3.00-3.00

PROBIT LOGIT

CD

F

Probit

Logit

LIMDEP : 73

Probit Logit LIMDEP

8.1 Probit

1 LIMDEP 3.1 18 20 Data Editor

2 Model/Binary Choice/Probit Main Menus

LIMDEP : 74

3 LIMDEP PROBIT 4

Dependent variable Menu button [ ] Y

Independent variables c 1 d X1, X2, D1, D2, D3 D4 ONE

5 LIMDEP Output

Command Bar Command Document LIMDEP Probit

PROBIT ; Lhs = [] ; Rhs = [] ; Option [] $

PROBIT ; Lhs=Y ; Rhs = ONE, X1, X2, D1, D2, D3, D4 $

LIMDEP Probit MLE Probit

1 2

LIMDEP : 76

=======================================================================

Analysis of Binary Choice Model Predictions Based on Threshold = .5000 -----------------------------------------------------------------------

Prediction Success -----------------------------------------------------------------------

Sensitivity = actual 1s correctly predicted 67.595% Specificity = actual 0s correctly predicted 51.213% Positive predictive value = predicted 1s that were actual 1s 59.598% Negative predictive value = predicted 0s that were actual 0s 59.748% Correct prediction = actual 1s and 0s correctly predicted 59.661% -----------------------------------------------------------------------

Prediction Failure -----------------------------------------------------------------------

False pos. for true neg. = actual 0s predicted as 1s 48.787% False neg. for true pos. = actual 1s predicted as 0s 32.405% False pos. for predicted pos. = predicted 1s actual 0s 40.402% False neg. for predicted neg. = predicted 0s actual 1s 40.252% False predictions = actual 1s and 0s incorrectly predicted 40.339% =======================================================================

LIMDEP 5

1 :

logL = The log likelihood function maximum logL0 = The log likelihood function all slopes = 0

(One) logL0 )]P1log()P1(PlogP[nLlog 0 += P sample proportion ones

Chi-square = H0: = 0 [] )LlogL(log2 0

2 = Degrees of freedom = Akaike Information Criterion (AIC) = n/)KL(log2 Bayesian Information Criterion (BIC) = n/)KlogKL(log2 Finite Sample AIC = n/))1Kn/()1K(KKL(log2 + HQIC = n/))nlog(logKL(log2

2 : 6 OLS

3 (Fit) : LIMDEP

P0 = Proportion 0

P1 = 1 P0 = y

LIMDEP : 77

LogL = =

+n1i

iiii )F1log()y1(Flogy Fi Predicted probability yi = 1 |

xi Predicted probability yi = (1-yi)(1-Fi) + yiFi

LogL0 = )PlogPPlogP(n 1100 + McFadden R2 = 1 LogL / LogL0

4 : 22 0 0 0 1 1 0 1 1 0 0 24.8% 1 1 34.9%

5 : 59.66% 1 67.60% 1 0 51.21% 0

Probit LIMDEP Options

Options LIMDEP

Marginal Effect = )x'(f

x)x|y(E ; Marginal Effects

Predict yi Predict yi = 1 if )x'(F i > P* ; Keep = Predict probabilities )x'(F i ; Prob = Residuals yy ; Res = Sample Selection

)x'()x'()x|1y(obPr

i

iii

==

)x'(1)x'()x|0y(obPr

i

iii

==

; Hold (IMR = )

Options 1 1 4 Probit 73 74

Output

LIMDEP : 78

2 Options

Sample Selection Keep results for sample selection model/Keep IMR as variable

Predict probabilities Keep probabilities as variable

Marginal Effect Display marginal effects

Predict yi Keep predictions as variable

Residuals Keep residuals as variable 3 Options

LIMDEP Marginal Effect Output Sample selection, Predict probabilities, Predict yi Residuals Data Editor

Command Bar Command Document Option Probit

PROBIT ; Lhs = [] ; Rhs = [] ; Margin ; Hold (IMR= []) ; Prob = []; Keep = [] ; Res = [] $

PROBIT ; Lhs=Y ; Rhs = ONE, X1, X2, D1, D2, D3, D4 ; Margin ; Hold (IMR=SS) ; Prob = Ypro ; Keep = Ypre ; Res = Res $

LIMDEP : 79

LIMDEP

+-------------------------------------------+

| Partial derivatives of E[y] = F[*] with | | respect to the vector of characteristics. | | They are computed at the means of the Xs. | | Observations used for means are All Obs. | +-------------------------------------------+

+---------+--------------+----------------+--------+---------+----------+

|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |Elasticity| +---------+--------------+----------------+--------+---------+----------+

Constant .01693818 .10616393 .160 .8732 X1 .00029450 .777897D-04 3.786 .0002 .45759914 X2 -.04953140 .01735118 -2.855 .0043 -.40809423 Marginal effect for dummy variable is P|1 - P|0. D1 -.11120875 .05959634 -1.866 .0620 -.19156018 Marginal effect for dummy variable is P|1 - P|0. D2 .10566744 .04645449 2.275 .0229 .10530684 Marginal effect for dummy variable is P|1 - P|0. D3 .09260099 .04115903 2.250 .0245 .05270081 Marginal effect for dummy variable is P|1 - P|0. D4 -.26184082 .09438318 -2.774 .0055 -.01457066

8.2 Logit

1 LIMDEP Model/Binary Choice/Logit Main Menus

Marginal Effect Probit

LIMDEP : 80

3 LIMDEP LOGIT (Binomial) 4

Dependent variable Menu button [ ] Y

1 2

LIMDEP : 81

Independent variables c 1 d X1, X2, D1, D2, D3 D4 ONE

5 LIMDEP Output

Command Bar Command Document LIMDEP Logit

LOGIT ; Lhs = [] ; Rhs = [] ; Option [] $

LOGIT ; Lhs = Y ; Rhs = ONE, X1, X2, D1, D2, D3, D4$

LIMDEP Logit MLE Logit +---------------------------------------------+

| Multinomial Logit Model | | Maximum Likelihood Estimates | | Model estimated: Oct 22, 2007 at 02:08:43PM.| | Dependent variable Y | | Weighting variable None | | Number of observations 766 | | Iterations completed 4 | | Log likelihood function -516.0052 | | Number of parameters 7 | | Akaike IC= 1046.010 Bayes IC= 1078.499 | | Finite sample corrected AIC = 1046.158 | | Restricted log likelihood -530.5747 | | Chi squared 29.13902 | | Degrees of freedom 6 | | Prob[ChiSqd > value] = .5725929E-04 | | Hosmer-Lemeshow chi-squared = 45.47601 | | P-value= .00000 with deg.fr. = 8 | +---------------------------------------------+

+---------+--------------+----------------+--------+---------+----------+


Characteristics in numerator of Prob[Y = 1] Constant .06884274 .42892877 .160 .8725 X1 .00119064 .00031636 3.764 .0002 801.959530 X2 -.20036646 .07052652 -2.841 .0045 4.25237859 D1 -.45773447 .24978521 -1.833 .0669 .88903394 D2 .42988696 .18979108 2.265 .0235 .51436031 D3 .37875239 .16936321 2.236 .0253 .29373368 D4 -1.15365849 .48370128 -2.385 .0171 .02872063

LIMDEP : 83

=======================================================================

Analysis of Binary Choice Model Predictions Based on Threshold = .5000 -----------------------------------------------------------------------

Prediction Success -----------------------------------------------------------------------

Sensitivity = actual 1s correctly predicted 67.595% Specificity = actual 0s correctly predicted 51.213% Positive predictive value = predicted 1s that were actual 1s 59.598% Negative predictive value = predicted 0s that were actual 0s 59.748% Correct prediction = actual 1s and 0s correctly predicted 59.661% -----------------------------------------------------------------------

Prediction Failure -----------------------------------------------------------------------

False pos. for true neg. = actual 0s predicted as 1s 48.787% False neg. for true pos. = actual 1s predicted as 0s 32.405% False pos. for predicted pos. = predicted 1s actual 0s 40.402% False neg. for predicted neg. = predicted 0s actual 1s 40.252% False predictions = actual 1s and 0s incorrectly predicted 40.339% =======================================================================

Logit Probit , Mc-Fadden R2 Options Probit

1 1 4 Logit 79 81 Output

2 Options Sample Selection Keep results for sample selection model Predict probabilities Keep probabilities as variable Marginal Effect Display marginal effects Predict yi Keep predictions as variable Residuals Keep residuals as variable

LIMDEP : 84

3 Options LIMDEP Marginal Effect Output Sample selection, Predict probabilities, Predict yi Residuals Data Editor

Command Bar Command Document Options Logit

LOGIT ; Lhs = [] ; Rhs = [] ; Margin ; Hold (IMR= []) ; Prob = []; Keep = [] ; Res = [] $

LOGIT ; Lhs=Y ; Rhs = ONE, X1, X2, D1, D2, D3, D4 ; Margin ; Hold (IMR=SS) ; Prob = Ypro ; Keep = Ypre ; Res = Res $

LIMDEP

Marginal Effect Logit

Marginal Effect Logit

LIMDEP : 85

Probit Logit Marginal Effect (X1), (D2) (D3) (X2), (D1) (D4)

9. Ordered Choice Models

(: Ordinal scale) , (Satisfaction rating) (Probability) Ordered Choice Models Binary Choice Models

1]x|[Var,0]x|[E),|(F~,x'y iiiiiiii*i ==+=

iy = 0 0iy , = 1 1i0 y

LIMDEP : 86

1J210

LIMDEP : 87

2 Model/Discrete Choice/Ordered Main Menus

3 LIMDEP ORDERED

1 2

LIMDEP : 88

4 Dependent variable Menu button [ ]

Y Independent variables c

1 d X1, X2, D1 D2 ONE

5 LIMDEP Output

Command Bar Command Document LIMDEP Ordered Probit

ORDERED ; Lhs = [] ; Rhs = [] ; Option [] $

ORDERED ; Lhs = Y ; Rhs = ONE, X1, X2, D1, D2 $

LIMDEP Ordered Probit MLE Ordered Probit +---------------------------------------------+

| Ordered Probability Model | | Maximum Likelihood Estimates | | Model estimated: Oct 25, 2007 at 04:24:52PM.| | Dependent variable Y | | Weighting variable None | | Number of observations 8140 | | Iterations completed 14 | | Log likelihood function -11284.69 | | Number of parameters 9 | | Akaike IC=22587.373 Bayes IC=22650.414 | | Finite sample corrected AIC =22587.395 | | Restricted log likelihood -11308.02 | | Chi squared 46.66728 | | Degrees of freedom 4 | | Prob[ChiSqd > value] = .0000000 | | Underlying probabilities based on Normal | | Cell frequencies for outcomes | | Y Count Freq Y Count Freq Y Count Freq | | 0 447 .054 1 255 .031 2 642 .078 | | 3 1173 .144 4 1390 .170 5 4233 .520 | +---------------------------------------------+

+---------+--------------+----------------+--------+---------+----------+


Index function for probability Constant 1.32892003 .07275667 18.265 .0000 X1 .35589972 .07831928 4.544 .0000 .32998942 X2 .00927670 .00629721 1.473 .1407 10.8759203 D1 .10603682 .02664775 3.979 .0001 .33169533 D2 .04525826 .02546350 1.777 .0755 .52936118

LIMDEP : 89

Threshold parameters for index Mu(1) .23634787 .01236704 19.111 .0000 Mu(2) .62954430 .01439990 43.719 .0000 Mu(3) 1.10763798 .01405938 78.783 .0000 Mu(4) 1.55676228 .01527126 101.941 .0000 +---------------------------------------------------------------------------+

| Cross tabulation of predictions. Row is actual, column is predicted. | | Model = Probit . Prediction is number of the most probable cell. | +-------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+

| Actual|Row Sum| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | +-------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+

| 0| 447| 0| 0| 0| 0| 0| 447| | 1| 255| 0| 0| 0| 0| 0| 255| | 2| 642| 0| 0| 0| 0| 0| 642| | 3| 1173| 0| 0| 0| 0| 0| 1173| | 4| 1390| 0| 0| 0| 0| 0| 1390| | 5| 4233| 0| 0| 0| 0| 0| 4233| +-------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+

|Col Sum| 8140| 0| 0| 0| 0| 0| 8140| 0| 0| 0| 0| +-------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+

LIMDEP 3

Probit Logit Cell frequencies

2 MLE Standard Error, t-statistic, P-value

3 Cross tab Row Column

Ordered Probit LIMDEP Options

Margianal Effects ; Marginal Effects

Predict yi ; Keep = []

Residuals ; Res = []

Predict probabilities ; Prob = []

Options 1 1 4 Oredered Probit 86 88

Output

LIMDEP : 90

2 Display marginal effects Marginal effects Fitted Values

3 Options Predict y Residuals

Predict yi Keep predictions as variable

Residuals Keep residuals as variable : Predict probabilities Options

Ordered Probit

4 Options LIMDEP Marginal Effect Output Predict yi Residuals Data Editor

LIMDEP : 91

Command Bar Command Document Options Ordered Probit

ORDERED ; Lhs = [] ; Rhs = [] ; Margin ; Prob = [] ; Keep = [] ; Res = [] $

ORDERED ; Lhs = Y ; Rhs = ONE, X1, X2, D1, D2 ; Margin ; Prob = Ypro; Keep = Ypre ; Res = Res $

LIMDEP

+----------------------------------------------------+

| Marginal effects for ordered probability model | | M.E.s for dummy variables are Pr[y|x=1]-Pr[y|x=0] | | Names for dummy variables are marked by *. | +----------------------------------------------------+

+---------+--------------+----------------+--------+---------+----------+


These are the effects on Prob[Y=00] at means. Constant .000000 ......(Fixed Parameter)....... X1 -.03907461 .00862973 -4.528 .0000 .32998942 X2 -.00101850 .00069179 -1.472 .1409 10.8759203 *D1 -.01131976 .00277405 -4.081 .0000 .33169533 *D2 -.00498024 .00280960 -1.773 .0763 .52936118 These are the effects on Prob[Y=01] at means. Constant .000000 ......(Fixed Parameter)....... X1 -.01647123 .00362630 -4.542 .0000 .32998942 X2 -.00042933 .00029148 -1.473 .1408 10.8759203 *D1 -.00483428 .00119623 -4.041 .0001 .33169533 *D2 -.00209669 .00118069 -1.776 .0758 .52936118 These are the effects on Prob[Y=02] at means. Constant .000000 ......(Fixed Parameter)....... X1 -.03256547 .00934016 -3.487 .0005 .32998942 X2 -.00084883 .00042174 -2.013 .0441 10.8759203 *D1 -.00963993 .00301864 -3.193 .0014 .33169533 *D2 -.00414232 .00217061 -1.908 .0563 .52936118

Marginal Effect Ordered Probit

LIMDEP : 92

These are the effects on Prob[Y=03] at means. Constant .000000 ......(Fixed Parameter)....... X1 -.03726702 .00947294 -3.934 .0001 .32998942 X2 -.00097138 .00083974 -1.157 .2474 10.8759203 *D1 -.01121164 .00413864 -2.709 .0067 .33169533 *D2 -.00473412 .00324171 -1.460 .1442 .52936118 These are the effects on Prob[Y=04] at means. Constant .000000 ......(Fixed Parameter)....... X1 -.01643041 .00658667 -2.494 .0126 .32998942 X2 -.00042827 .00017168 -2.494 .0126 10.8759203 *D1 -.00518111 .00245201 -2.113 .0346 .33169533 *D2 -.00207949 .00135601 -1.534 .1251 .52936118 These are the effects on Prob[Y=05] at means. Constant .000000 ......(Fixed Parameter)....... X1 .14180874 .03108209 4.562 .0000 .32998942 X2 .00369631 .140476D-04 263.129 .0000 10.8759203 *D1 .04218672 .00021778 193.710 .0000 .33169533 *D2 .01803286 .01015706 1.775 .0758 .52936118 +-------------------------------------------------------------------------+

| Summary of Marginal Effects for Ordered Probability Model (probit) | +-------------------------------------------------------------------------+

Variable| Y=00 Y=01 Y=02 Y=03 Y=04 Y=05 Y=06 Y=07 | --------------------------------------------------------------------------+

ONE .0000 .0000 .0000 .0000 .0000 .0000 X1 -.0391 -.0165 -.0326 -.0373 -.0164 .1418 X2 -.0010 -.0004 -.0008 -.0010 -.0004 .0037 *D1 -.0113 -.0048 -.0096 -.0112 -.0052 .0422 *D2 -.0050 -.0021 -.0041 -.0047 -.0021 .0180

9.2 Ordered Logit

1 1 4 Ordered Probit 86 88 Options

2 Model: Logit LIMDEP

Output

LIMDEP : 93

Command Bar Command Document Option Ordered Logit Ordered Probit

ORDERED ; Lhs = [] ; Rhs = [] ; LOGIT $

ORDERED ; Lhs = Y ; Rhs = ONE, X1, X2, D1, D2 ; LOGIT $

LIMDEP Ordered Logit MLE Ordered Logit +---------------------------------------------+

| Ordered Probability Model | | Maximum Likelihood Estimates | | Model estimated: Oct 25, 2007 at 10:16:11PM.| | Dependent variable Y | | Weighting variable None | | Number of observations 8140 | | Iterations completed 14 | | Log likelihood function -11288.59 | | Number of parameters 9 | | Akaike IC=22595.181 Bayes IC=22658.222 | | Finite sample corrected AIC =22595.203 | | Restricted log likelihood -11308.02 | | Chi squared 38.85922 | | Degrees of freedom 4 | | Prob[ChiSqd > value] = .0000000 | | Underlying probabilities based on Logistic | | Cell frequencies for outcomes | | Y Count Freq Y Count Freq Y Count Freq | | 0 447 .054 1 255 .031 2 642 .078 | | 3 1173 .144 4 1390 .170 5 4233 .520 | +---------------------------------------------+

Ordered Logit +---------+--------------+----------------+--------+---------+----------+


Index function for probability Constant 2.47112743 .12111244 20.404 .0000 X1 .54023638 .13258707 4.075 .0000 .3299

Documents

Handbook Limdep