Overview: LIMDEP 8.0

LIMDEP takes the form of an econometrics studio. Analysis of a data set is done

interactively in a set of windows. Program control may be from a 'script' or in

an unstructured session of instructions and manipulations. The program is

designed to allow easy setup of data for estimation, specification of different

forms of the models, experimentation with different specifications, hypothesis

testing, analysis of data and model results and construction of special procedures

and estimators. The following sections will list in detail the set of program

features and tools supported in Version 8.0.

CROSS SECTION, PANEL DATA, and TIME SERIES:

CROSS SECTION

Most of the models and data analyses done with LIMDEP are oriented to cross

sectional, individual level data. These include:

• Descriptive statistics
• Descriptive graphics
• Linear regression models
• Models for discrete choices
• Models for censoring and truncation
• Models for count data
• Models for stochastic frontier production and cost functions
• Models for survival and duration data
• Numeric, alphanumeric (string), and date series; value labels
• Extensive library of operators and statistical, mathematical, date and string functions
• and many more. Data sets in this context may be large. The program allows up to 3,000,000 observations and 900 variables, which should accommodate all but the most ambitious analyses.

PANEL DATA

Nearly all of the models in LIMDEP may be analyzed with special tools for panel data. This includes fixed and random effects, random parameters and latent class models for almost all nonlinear models supported by the package. There are also numerous special estimators for the linear model, such as Arellano and Bond's GMM estimator for dynamic panels and Hausman and Taylor's estimator for random effects models. No panel data operation anywhere in the program requires that the data set be balanced. Most estimators place no limit on the number of groups in the panel - the data set is already 'in the program' so it must already fit in memory. Many tools in addition to the estimation programs are also provided. For example, you can bootstrap sample groups in your panel data set, a feature we have not seen anywhere else.

TIME SERIES

LIMDEP is not best known for its time series features. Nonetheless, a range of estimators for time series modeling are presented, including autoregressive and moving average models, ARMAX models, GARCH and GARCH-in-mean models. We also provide a set of analysis tools, such as spectral density estimation, ACF and PACF, Phillips-Perron tests, the Newey-West estimator, and so on.

MODELS:

A full list of the models that can be fit with LIMDEP appears elsewhere so we'll just give a partial list here:

CROSS SECTION

• Descriptive statistics including means, variance, skewness, etc.
• Linear regression
• Multiple equations linear and nonlinear models
• Loglinear models: Weibull, inverse Gauss, beta, log-gamma
• Binary choice including probit, logit, and several other formulations
• Bivariate probit
• Multivariate probit
• Logit models, binomial and multinomial - discrete choice models
• Ordered probit and logit
• Nonlinear least squares regression
• Censored data (tobit)
• Models for count data
• Survival models
• Sample selection
• Stochastic frontiers.

PANEL DATA

• Fixed effects (true fixed effects - the dummy variable coefficients are estimated)
• Time series - cross section, variance structure models
• Random effects
• Random parameters
• Latent class
• Dynamic panel data models - Arelland-Bond-Bover estimator
• Hausman and Taylor's instrumental variable estimator

TIME SERIES MODELS

·        ARMAX

·        GARCH and GARCH in mean

·        Autoregressive and moving average regressions

USER SPECIFIED MODELS

MAXIMIZE/MINIMIZE allow you to specify the likelihood function for your own model. GMM estimation can be specified instead.

POST ESTIMATION:

Model estimation is only part of the data analysis. By 'post estimation,' we mean the manipulation of model results along with other statistics and procedures.

RECOVERABLE RESULTS

All model estimates are 'recoverable.' Coefficients and asymptotic covariance matrices are retained and integrated into the matrix algebra package. Numerical values such as log likelihoods and sums of squares are recoverable and useable, by name, for example, in testing hypotheses or computing diagnostic statistics.

SIMULATION AND PREDICTION

Estimated models may be used for simulations and for computing predictions, generalized residuals, and other functions for forecasting and specification analysis.

PROGRAMMING TOOLS FOR MANIPULATING MODEL RESULTS

Program tools are provided for manipulating results. One of the most useful is the WALD command which is used to compute nonlinear functions and asymptotic variances for nonlinear functions of model estimates. Programming the derivatives for the delta method is unnecessary.

ESTIMATION EXTENSIONS

Facilities are provided for bootstrap sampling cross sections and from panels. Thus, bootstrap standard errors can easily be computed. All model results can be retained for use in other program functions. Programming two step estimators is simple.

PROGRAMMING TOOLS AND FLEXIBILITY:

LIMDEP's program features are an extensive set of fully integrated analysis tools. All of these tools interact to produce a studio, or 'laboratory' format for econometric analysis. Exploratory manipulation and estimation, including experimentation such as Monte Carlo analysis, are straightforward. These tools include:

DATA INPUT AND OUTPUT

Read and write ASCII, binary, spreadsheet, and other types of files.

TRANSFORMATIONS

Algebraic transformations, dozens of functions, sort, recode, expand and collapse categorical variables.

SAMPLE DEFINITION

Include or reject by algebraic criteria, time periods by date, bootstrapping cross sections or panel data, random sampling with random number generators.

MATRIX ALGEBRA

Full algebraic capabilities and over 75 matrix functions such as determinant, rank, characteristic roots, several types of inverses, and so on. Data and model results are combined seamlessly in this routine.

SCIENTIFIC CALCULATOR

Over 100 functions plus algebraic results.

ESTIMATION PROGRAMS

Nearly 100 different preprogrammed estimators for all types of cross section, time series, and panel data settings, including continuous data, discrete choice, count data, survival data, and censored, truncated and limited dependent variables.

USER DEFINED MODELS

MAXIMIZE/MINIMZE allow you to define your own optimization problem, including maximum likelihood, GMM and nonlinear least squares. These also allow quadrature and simulation estimators.

PROGRAMMING TOOLS

Procedures with adjustable parameters, DO FOR, DO WHILE and DO UNTIL, EXECUTE with bootstrapping and so on allow you to write your own iterative procedures

NUMERICAL ANALYSIS

Integration, differentiation, function plotting, computation of variances for nonlinear functions of estimates.

DOCUMENTATION:

LIMDEP 8.0 DOCUMENTATION

The LIMDEP 8.0 set of manuals, with nearly 2000 pages, contains full reference guides for the program, background econometrics, and sample applications. The LIMDEP documentation consists of two parts:

LIMDEP 8.0 Reference Guide

The Reference Guide provides all instructions for operating the program, including installation, invocation, and most of the basic setup operations that precede model estimation. These operations include reading and transforming data and setting the sample. This manual also describes the optimization procedures, how to use the matrix algebra package and scalar scientific calculator as stand alone tools and as part of LIMDEP programs, what sorts of results are produced by the program, and some of the common features of the model estimation programs, such as how to do post estimation analysis of model results. Two other components of the Reference Guide are a summary of how the model commands are documented in the Econometric Modeling Guide and a complete listing of the program diagnostics.

LIMDEP 8.0 Econometric Modeling Guide, Volume 1 and Volume 2

The 29 chapters of this guide are arranged in two volumes. These provide the econometric background, LIMDEP commands, and examples with data, commands and results. Topics are arranged by modeling framework, not by program command. There are chapters on

·          Descriptive statistics

·        Linear regression

·        Panel data analysis

·        Heteroscedasticity

·        Binary choice models

·        Models for count data

·        Censored and truncated data

·        Survival models

and many others. Each model fit by the program is fully documented. The full set of formulas for all computations are shown in this manual with the full mathematical documentation of the models. Additional chapters in this guide show how to do numerical analysis, how to program your own estimators, and provide a full listing of diagnostics.

NLOGIT 3.0 DOCUMENTATION

The NLOGIT 3.0 set of manuals consist of the three LIMDEP 8.0 manuals plus a separate reference guide for NLOGIT.

NLOGIT 3.0 Reference Guide

The roughly 200 pages of this NLOGIT Reference Guide give complete instructions for specifying and estimating discrete choice models with NLOGIT. The model simulator is also documented here.The LIMDEP 8.0 set of manuals, with nearly 2000 pages, contains full reference guides for the program, background econometrics, and sample applications. The LIMDEP documentation consists of two parts:

ACCURACY:

LIMDEP uses extremely accurate computational methods throughout. The National Institute of Standards and Technology (NIST) has published a suite of about 50 benchmark tests with certified solutions for assessing program accuracy in computing descriptive statistics, analysis of variance, linear regression and nonlinear least squares regression. LIMDEP scores high marks on all of these (see B.D. McCullough's review in the March, 1999 issue of Journal of Applied Econometrics.).

PROGRAM CAPABILITIES IN DETAIL

LIMDEP contains extensive sets of tools for every step in the analysis of a data set.

DATA MANAGEMENT

Data setup includes input, transformation, and setting sample specifications. Data management tools provide for input of data or internal generation with the random number generators, and other preparation of data for use in model estimation and analysis. Also sample definition is specified by selection of observations or bootstrapping random samples from within the data.

DATA INPUT AND OUTPUT

These are features of the program operation of importing data into the program and exporting data from LIMDEP to other programs - the read (import) and write (export) operations. You can also transform and examine the data within the program.

Read (import data)

·          ASCII, XLS, WKS, Binary, DIF, CSV

·          LIMDEP is supported by DBMS/Copy and Stat/Transfer

·          Merge individual and group level data in a panel data set

·          Recode alphanumeric data to numeric values on input

·          Observation labels in spreadsheet files

·          Up to 3,000,000 observations

·          Blanks and other nonnumerics automatically converted to missing values

·          SAS ‘.’ automatically converted to missing values

·          Merge data with command files for scripts

·          Fortran formatted

·          Append observations to existing variables (stack multiple files)

·          Format by observations (rows) or by variables (columns) - transpose

Write (export data)

·        ASCII

·        CSV (can be read directly into Excel, Lotus)

·        DIF

·        Binary

·        WKS

·        LIMDEP system files can be read by Stat/Transfer and DBMS/Copy

·        Matrices may be written as exported data files

·         

Spreadsheet style data editor

Transformations: algebraic, recoding, sorting, existing or new variables

Random number generators for random sampling

List observations

Sample selection and definition: include, reject, sample

Read from and write to output window on the screen

DATA TRANSFORMATIONS

These operations are used to compute functions of your data and to create new variables. Data may either be imported into the environment or may be created internally using the random number generators.

Command Structures

·        Conditional: If(...), Else, Then...

·        Parentheses to any number of levels

·        Implied multiplication of grouped variables and expressions

Algebraic Transformations

·          Standard operators: + - * / ^ (power)

·          Binary variable operators: x > 1 = 1(x > 1) likewise for <, >= etc.

·          Comparison operators: x ! y = max(x,y), x ~ y = ,in(x,y)

·          Expand categorical variable into a set of dummy variables

Functions

·          Log, Exp, Abs, Sqr, Sin, Rsn (arcsin), Cos, Rcs (arccos), Tan

·          Gamma, digamma, trigamma, log gamma, beta

·          Sign, fix (round to nearest integer), integer part

·          Box-Cox transformation and derivatives of Box-Cox

Probability Distributions

·          Logistic: logit function, density, cdf

·          Univariate normal: density, cdf, truncated means and variances, sample selection ‘lambda,’ inverse normal cdf, inverse cdf to pdf

·          Bivariate normal: pdf, cdf, partial derivatives

·          Multivariate normal: cdf

Trends and Seasonal Dummy Variables

Stratification Variables and Period Variables for Panels

Leads and Lags

Matrix Functions: Dot Products, quadratic Forms

Sample Statistical Transformations

·        Means, deviations, standardized variables

·        Panel data, group means and deviations

·        Rowwise moments of a set of variables

·        Multiple of a set of variables

Random Number Generation

·        Continuous random variables, uniform, normal, lognormal, t, chi-squared, F, exponential, Weibull, Gumbel, gamma, beta, logistic, Cauchy, truncated standard normal

·        Discrete random variables: Poisson, discrete uniform, binomial, geometric

·        Halton sequences

Recode Ranges of Values or Recode Specific Values to Discrete

Sort Ascending or Descending (Carry other Variables and/or Labels)

SAMPLING AND BOOTSTRAPPING

These features are used to define the specific observations to be used in the estimation procedures or other computations that access the data. The first group are used to define the ‘current sample.’ The second group are used to ‘subsample’ from the sample, using either a random subset of the current sample, or the ‘leave one out’ procedure of the jackknife procedure.

Sampling

·          Include observations based on algebraic condition

·          Reject observations based on algebraic conditions

·          Period - specify time interval to be in sample

·          Sample - specify particular observations or ranges of observations

Bootstrap and Jacknife

·        Draw specified number of observations from current sample

·        Draw with replacement

·        Draw specified number of groups in a panel data set

·        Execute a procedure a specified number of times, drawing a new bootstrap sample with each repetition

·        Execute using jackknife procedure for sampling

·        Maximum score binary choice estimator uses bootstrapping

·        Regression, least absolute deviations uses bootstrapping

The EXECUTE procedure may be used to bootstrap any estimator in the program whether one of the supported procedures or one that is created by the user. The command specifies the matrix or scalar to be bootstrapped. It may be anything that the program specifies.

Application

A simple application illustrates estimating a standard error for a nonparametric statistic. Suppose we have two sets of rankings of 100 items, labeled R1 and R2. The rank correlation between them is computed as

Since the statistic is distribution free, we can only approximate its asymptotic variance. The following computes 100 bootstrap samples and estimates the mean, variance, skewness and kurtosis of W and displays a histogram of the estimates.

PROC

CREATE  ; DiffSq = (R1 - R2)^2 $

CALC    ; RankCor = 1 - 6 * Sum(DiffSq) / (n*(n^2-1)) $

ENDPROC $

EXECUTE ; N = 100 ; Bootstrap = RankCor ; Histogram $

Completed   100 bootstrap iterations.

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

| Results of bootstrap estimation of model.|

| Model has been reestimated   100 times.  |

| Statistics shown below are centered      |

| around the  original estimate  based on  |

| the original full sample of observations.|

| Result is RANKCOR  =       .34512        |

| Bootstrap samples have  100 observations.|

| Estimate  RtMnSqDev  Skewness   Kurtosis |

|     .345       .180     1.278      1.790 |

| Minimum =      .254  Maximum =      .701 |

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

Features02

The bootstrapped quantity in the procedure can be anything that the program computes using any instruction, model estimator, or other procedure.

MONTE CARLO ANALYSIS

Random Number Generation

·          Continuous random variables: uniform, normal, lognormal, t, chi-squared, F, exponential, Weibull, Gumbel, gamma, beta, logistic, Cauchy, truncated standard normal

·          Discrete random variables: Poisson, discrete uniform, binomial, geometric

·          Several estimators use Halton draws instead (pseudo Monte Carlo draws). Users may generate Halton sequences.

Bootstrapping

·        Draw specified number of observations from current sample

·        Draw with replacement

·        Draw specified number of groups in a panel data set

·        Execute a procedure a specified number of times, drawing a new bootstrap sample with each repetition

The executed procedure may be used to bootstrap any estimator in the program whether one of the supported procedures or one that is created by the user. The command specifies the matrix or scalar to be bootstrapped. It may be anything that the program specifies.

Estimation

Randomly generated data enter the sample exactly the same as external, real data. Monte Carlo samples may be specified and used with any estimation procedure or any other part of the program that uses data.

Example: Fixed effects probit model

The following program does a Monte Carlo study of the fixed effects probit estimator. We fit the model 20 times with randomly generated data and compute descriptive statistics for the estimated parameter vectors. The underlying theory states that the estimator is biased when T is small. (The example is consistent with it - the estimator appears to be badly biased in our study with T = 5.) This program illustrates several features of LIMDEP.

/*  We investigate the behavior of the fixed effects estimator

    in the panel probit model

    y*(i,t) = a(i) + b1 * x(i,t) + b2 * d(i,t) + e(i,t),

    x(i,t) is a continuous variable, d(i,t) is a dummy variable

    and both b1 and b2 equal 1.  The model is fit 20 times

    with randomly generated data on a(i) x(i,t) and d(i,t) all

    (reused) and e(i,t) resampled for each replication.

*/

SAMPLE ; 1 - 5000 $ (Sample length is constant)

CREATE ; xit = rnn(0,1) ; dit=(rnu(-.5,.5)+xit) > 0 

       ; i5=trn(5,0) $  (I5 = 1,1,1,1,1,2,2,2,2,2,...)

MATRIX ; u=rndm(1000) ?  Random matrix for fixed effects

       ; a5  = gxbr(xit,i5)$ Group means of x variable

CREATE ; ai = sqr(5)*a5(i5)+u(i5) ? Fixed effects correlated with x

       ; index  =  ai + xit + dit ; br = 0 ; dr = 0 $

NAMELIST; repl = br,dr $      ? Group two variables

?

? Achieve replicability by using a known seed for generator

?

CALC   ; Ran(55377) $

?

? Set parameters for the simulation. Values can be changed

?

PROC  = MCset(Tir,Nir) $

CALC    ; Ti = Tir ; Nobs = Nir ; NT = Ti*Nobs ; i = 0 $

SAMPLE  ; 1 - NT $  Sample set to NT = 5000.

MATRIX  ; beta = init(20,2,0.0) $  Place to store results

ENDPROC

?

? This runs the replication.

? Binomial Probit is fit by unconditional maximum likelihood

?

PROC = MCrun$

CREATE ; eit = Rnn(0,1) ; yit = (index + eit) > 0 $

CALC   ; i = i + 1 $

PROBIT ; Lhs = yit ; Rhs = xit,dit ;  Pds = Ti ; Fixed effects $

MATRIX ; beta(i,*) = b $   Save estimated parameters in row i.

ENDPROC

?

? T=5, N=1000 groups

?

EXECUTE ; Proc = MCset(5,1000) $            ? Setup simulation

EXECUTE ; Proc = MCrun ; n = 20 ; Silent $  ? 20 replications

SAMPLE  ; 1 - 20 $                          ? Describe results

CREATE  ; repl = beta $       ? Copy matrix to 2 variables

DSTAT   ; rhs  = repl $       ? Examine the results

The following results emerge. The theory appears to be correct; the estimates (means) deviate substantially from the theoretical values of 1.0 and 1.0.

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

Variable        Mean         Std.Dev.        Minimum         Maximum

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

BR        1.60317327      .100071915      1.37897075      1.82531768

DR        1.42251625      .190029262      1.04402871      1.78544870

WEIGHTED DATA

Weighted Least Squares and Weighted Log Likelihoods

Weights may be used with any estimator in any context.

Replication Weights

Weights are automatically scaled to sum to the number of observations in the sample. You may override this to request replications or simple unscaled weights.

Panel Data

Time invariant weights may be used to weight groups in panel data estimators.

Choice Based Sampling

Endogenous sampling weights may be specified for discrete choice models. This is coupled with a robust estimator of the asymptotic covariance matrix of the estimator. (Manski-Lerman WESML estimator.)

RANDOM NUMBER GENERATION

Stata of the Art Random Number Generator

LIMDEP uses L’Ecuyer’s multiple part random number generator. This generator has a period of roughly 2¹³² draws. (This is more than 10³⁹.) This is large enough to avoid recycling in any conceivable Monte Carlo study.

·          Set seed for the generator: random samples can be replicated.

·          Continuous random variables: uniform, normal, truncated normal, lognormal, t, chi-squared, F, exponential, Weibull, Gumbel, gamma, beta, logistic, Cauchy

·          Discrete random variables: Poisson, discrete uniform, binomial, geometric

Randomness Tests

The random number generators in LIMDEP and three other programs were subjected to the ‘DIEHARD’ battery of 15 tests of randomness in McCullough’s (Journal of Applied Econometrics, 1999) study. LIMDEP’s generator passed all but one. Only one of the four programs passed all tests and the other two failed most of them.

Halton Sequences

This is a new method of computing simulation based integrals through use of nonrandom sequences. LIMDEP uses these in several places, and provides a function so users can create them for their own estimators.

DATA DESCRIPTION AND GRAPHICS

Descriptive statistics and graphical tools provide a range of options for preliminary description of your data set. Data description includes computation of statistics as well as graphical tools such as scatter plots and time plots.

DESCRIPTIVE STATISTICS FOR CROSS SECTIONS AND PANELS

Summary Measures

·          Means (arithmetic, geometric), standard deviations, minima, maxima

·          Medians, sample quantiles (deciles, quartiles)

·          Covariances

·          Correlations (Pearson, rank)

·          Coefficient of concordance for a set of ranks

·          Autocorrelations

·          Canonical correlations

·          Principal components

·          Condition number for data matrices

Normality Test

·          Skewness, kurtosis

·          Normal-quantile plot

·          Chi-squared test

Example:

This is a description of Mroz’s (1987) Labor Supply Data. FAMINC is family income, stratified by KIDS, which indicates whether there are children in the household (1 = no, 2 = yes).

All results based on nonmissing observations.

Stratification is based on KIDS

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

Variable        Mean       Std.Dev.      Minimum       Maximum    Cases

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

Stratum is KIDS     =    1.000.  Obs.=    79.000, Sum of wts. =  79.000

FAMINC    21.2270000    11.1699737    3.30500000    63.2000000       79

--------Skewness=  1.3629   Kurtosis=  5.3140  Chisq=   42.08 Prob= .00

Stratum is KIDS     =    2.000.  Obs.=   171.000, Sum of wts. = 171.000

FAMINC    23.9105380    13.6056289    4.00000000    91.0440000      171

--------Skewness=  2.2943   Kurtosis= 10.7256  Chisq=  575.28 Prob= .00

All observations in current sample

FAMINC    23.0625400    12.9239815    3.30500000    91.0440000      250

--------Skewness=  2.1517   Kurtosis= 10.2941  Chisq=  747.12 Prob= .00

              Order Statistics for Variables

Percentiles for FAMINC

Min.        3.3050    10th       10.195         20th          13.783

25th        15.500    30th       16.235         40th          18.900

Med.        21.136    60th       23.225         70th          26.020

75th        27.925    80th       29.100         90th          36.175

Max.        91.044

Data Arrangements

·          Stratified data

·          Weights

Analysis of Variance

·          Balanced and unbalanced panels

Panel and stratified data:

·          Analysis of variance

·          F test for effects

Crosstabs

·        By row, column, total proportions

·        Independence test

·          Individual, or frequency data

·          Crosstabs saved as matrices

This is Agresti's example of cross classified data on political affiliation
(Analysis of Categorical Data).

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

|Cross Tabulation                                                 |

|Row variable is NIJ      (Out of range 0-49:      0)             |

|Number of Rows =  3      (NIJ      =  0 to  2)                   |

|Col variable is NIJ      (Out of range 0-49:      0)             |

|Number of Cols =  3      (NIJ      =  0 to  2)                   |

|Chi-squared independence tests:                                  |

|Chi-squared[   4] =  102.04903   Prob value =  .00000            |

|G-squared  [   4] =  105.66216   Prob value =  .00000            |

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

|Joint Frequencies for Row Variable NIJ       Column Variable NIJ |

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

|NIJ     | Total  |DEMOCR INDPND REBUBL                           |

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

|LIBERAL |   399  |   143    156    100                           |

|MODERATE|   470  |   119    210    141                           |

|CONSRVTV|   214  |    15     72    127                           |

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

|   Total|  1083  |   277    438    368                           |

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

Embedded results - transfer to other programs

Program results such as descriptive statistics are displayed in embedded windows that may be transported to other programs.

Accurate computations

This is one of the National Institute of Standards and Technology test examples for benchmarking the accuracy of descriptive statistics computations.

Dataset Name:  Maryland Pick-3 Lottery

Description:   This is an observed/"real world" data set

               consisting of 218 Maryland Pick-3 Lottery values

               from September 3, 1989 to April 14, 1990 (32 weeks).

               One 3-digit random number (from 000 to 999)

               is drawn per day, 7 days per week for most

               weeks, but fewer days per week for some weeks.

               We here use this data to test accuracy

               in summary statistics calculations.

Stat Category: Univariate: Summary Statistics

Reference:     None

Data:          "Real World"

               1    Response          : y = 3-digit random number

               0    Predictors

               218  Observations

Model:         Lower Level of Difficulty

               2    Parameters        : mu, sigma

               1    Response Variable : y

               0    Predictor Variables

               y    = mu + e

                                                  Certified Values

Sample Mean                                ybar:  518.958715596330

Sample Standard Deviation (denom. = n-1)      s:  291.699727470969

Sample Autocorrelation Coefficient (lag 1) r(1):  -0.120948622967393

Number of Observations:                             218

Data: Y

READ ; Nobs = 218 ; Nvar = 1 ; Names = y ; ByVariable $

162 671 933 414 788 730 817  33 536 875 670 236 473 167 877 980 316 950

456  92 517 557 956 954 104 178 794 278 147 773 437 435 502 610 582 780

689 562 964 791  28  97 848 281 858 538 660 972 671 613 867 448 738 966

139 636 847 659 754 243 122 455 195 968 793  59 730 361 574 522  97 762

431 158 429 414  22 629 788 999 187 215 810 782  47  34 108 986  25 644

829 630 315 567 919 331 207 412 242 607 668 944 749 168 864 442 533 805

372  63 458 777 416 340 436 140 919 350 510 572 905 900  85 389 473 758

444 169 625 692 140 897 672 288 312 860 724 226 884 508 976 741 476 417

831  15 318 432 241 114 799 955 833 358 935 146 630 830 440 642 356 373

271 715 367 393 190 669   8 861 108 795 269 590 326 866  64 523 862 840

219 382 998   4 628 305 747 247  34 747 729 645 856 974  24 568 24  694

608 480 410 729 947 293  53 930 223 203 677 227  62 455 387 318 562 242

428 968

DSTAT ; Rhs = y ; AR1 $

Descriptive Statistics

All results based on nonmissing observations.

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

Variable      Mean       Std.Dev.      Minimum       Maximum    Cases

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

Y       518.958716    291.699727    4.00000000    999.000000      218

Autocorrelation -.120948623

DESCRIPTIVE STATISTICS AND TOOLS FOR TIME SERIES

Statistics

·          Autocorrelation

·          Partial autocorrelation

·          Box-Pierce and Box-Ljung tests

·          Phillips-Perron unit root tests

·          Spectral density estimator

Regression

·          ARMA and ARMAX regression

·          GARCH and GARCH in mean

·          Durbin-Watson test

·          Godfrey/Breusch-Pagan test

·          CUSUM and CUSUM squares tests

·          Autocorrelation: Prais-Winsten, maximum likelihood

·          AR(p) regression

·          Hatanaka estimator

Graphics

·        Line plots

·        Multiple time series plots

·        Plot forecasts from ARMAX models

·        CUSUM plot

·        Time series dating

GRAPHICS TOOLS

Scatter diagrams

·        Include linear regression line

Features03
This is a plot of generalized cost against travel time from a study of
transportation mode choice.

·          Line styles

·          Specify ranges on axes

·          Titles and labels

Histograms

·        Discrete or continuous data

·        Specify bins, proportions or intervals

Features04

Graphical displays

·          Kernel density plot

·          Matrix of scatter plots

·          Line plots for time series

·          Multiple time series plots

·          Box and whisker plots

·          Residual plots (against observation, time, or other variables)

·          User specified function plots

·          Plot predictions from time series models

·          Normal quantile plots

·          Survival and hazard function plots 

MODEL ESTIMATION AND ANALYSIS

Well over 100 model formulations for continuous, discrete, limited and censored dependent variables are provided. All model frameworks provide a wide variety of different variants. For example, there are well over 20 different types of models for count data.

This section lists the packaged estimations such as regression, binary choice and so on. See the Programming list for features provided for user defined estimation programs.

LINEAR REGRESSION MODELS

Least Squares Regression

·          Least squares

·          Instrumental variables and 2SLS

·          Least absolute deviations with bootstrapped standard errors

·          Forward stepwise regression

·          Extreme accuracy

o         Condition number and variance inflation factors

o         Leverage values

Summary Statistics

·          Fit measures, F, R2, adjusted R2, sum of squares

·          Information criteria

·          Likelihood function

·          Durbin-Watson

·          Condition number for data matrix

Predictions and Residuals

·          List, plot, retain

·          Standardized residuals

·          Influence analysis, leverage values, diagonals of ‘hat’ matrix

·          Confidence interval for predictions

·          Fill missing values 

Robust estimation

·          White and heteroscedasticity adjusted covariance matrix

·          Newey-West estimator

·          Cluster corrected covariance matrix

·          Least absolute deviations - bootstrapped covariance matrix

Example:

Linear regression of generalized travel cost on travel time and in vehicle cost with predictions and residual plot.

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

| Ordinary    least squares regression    Weighting variable = none     |

| Dep. var. = GC       Mean=   110.8797619    , S.D.=   47.97835298     |

| Model size: Observations =     840, Parameters =   3, Deg.Fr.=    837 |

| Residuals:  Sum of squares= 1337515.612    , Std.Dev.=       39.97484 |

| Fit:        R-squared=  .307458, Adjusted R-squared =          .30580 |

| Model test: F[  2,    837] =  185.80,    Prob value =          .00000 |

| Diagnostic: Log-L =  -4288.5358, Restricted(b=0) Log-L =   -4442.8380 |

|             LogAmemiyaPrCrt.=    7.380, Akaike Info. Crt.=     10.218 |

| Autocorrel: Durbin-Watson Statistic =   1.10958,   Rho =       .44521 |

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

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

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

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

 Constant     79.36853490       2.6183166   30.313   .0000

 TTME        -.4247518299   .67928627E-01   -6.253   .0000     34.589286

 INVC         .9673869016   .52353170E-01   18.478   .0000     47.760714

       Obs.    Observed Y   Predicted Y       Residual   95% Forecast Interval

        1          70.000        93.854       -23.8541      30.3128   157.3953

        2          71.000        79.786        -8.7862      18.4510   141.1214

        3          70.000        72.507        -2.5075      10.8824   134.1326

        4          30.000        73.123       -43.1232       8.4745   137.7718

        5          68.000        95.233       -27.2329      32.3630   158.1028

        6          84.000        74.770         9.2305      13.0538   136.4852

        7          85.000        63.477        21.5225        .0582   126.8968

        8          50.000        74.253       -24.2527       9.6096   138.8957

        9          129.00        157.11       -28.1064      85.6257   228.5871

       10          195.00        155.46        39.5369      83.6985   227.2277

Features06

Specification tests

·          CUSUM

·          Omitted variables

·          Structural change

·          J, Cox, PE tests

Restrictions

·          F and Wald tests for linear restrictions

·          Restricted regression

·          Inequality restricted regression

·          Wald tests for nonlinear restrictions

·          Lagrange multiplier, likelihood ratio tests

Heteroscedasticity

·          Weighted least squares

·          Multiplicative heteroscedasticity, maximum likelihood

·          Goldfeld-Quandt, Breusch-Pagan tests

·          Groupwise heteroscedasticity

·          Heteroscedastic fixed and random effects

·          ARCH, GARCH, GARCH in mean

Autocorrelation

·          Durbin-Watson, Godfrey tests

·          ML, Prais-Winsten, Cochrane-Orcutt, Hildreth-Lu estimators

·          Hatanaka estimator for lagged dependent variable, 2SLS

·          Higher order autoregressive

Panel data

·          Analysis of variance and covariance

·          Fixed effects

·          Random effects

·          Random parameters (GLS, hierarchical)

·          Balanced or unbalanced panels

·          Autocorrelation correction

·          Heteroscedasticity and autocorrelation tests

·          LM and Hausman tests for effects

·          White and Newey-West robust estimators

·          Dynamic linear models (Arellano/Bond)

Systems of linear equations

·          2SLS

·          3SLS

·          Seemingly unrelated regressions

o         Autocorrelation

o         Heteroscedasticity

o         Singular equation systems with constraints

o         GLS and maximum likelihood

·          Cross and within equation constraints

·          Covariance structures

o         OLS, GLS

o         Panel corrected standard errors

o         Grouping of observation units

o         Autocorrelation

ROBUST, SEMIPARAMETRIC AND NONPARAMETRIC ESTIMATION

Non- and semiparametric estimators

·          Least absolute deviations linear regression

·          Maximum score for binary choice

·          Klein and Spady estimator for binary choice

·          Nonparametric, kernel density regression

Robust estimators

·          GMM estimation for user specified models

·          Kernel density estimation

·          Spectral density estimation

·          Random parameters models

·          Kernel weights for estimation

Robust covariance matrix estimators

·          Bootstrapping standard errors for any estimator

·          White and heteroscedasticity corrected estimators

·          Newey-West estimators

·          Choice based sampling discrete choice estimators

·          Cluster based asymptotic covariance matrices

Robust tests

·          Rank correlation

·          Coefficient of concordance

·          CUSUM test

·          Kolmogorov-Smirnov

·          Normality test - chi-squared

·          Box-Pierce and Box-Ljung

NONLINEAR AND LOGLINEAR REGRESSION MODELS

BINARY CHOICE MODELS

ORDERED CHOICE MODELS

MULTINOMINAL CHOICE MODELS

CENSORING AND TRUNCATION

SAMPLE SELECTION MODELS

MODELS FOR COUNT DATA

STOCHASTIC FRONTIER MODELS

SURVIVAL MODELS

TIME SERIES MODELS

PANEL DATA MODELS

STATISTICAL ANALYSIS AND MODELING TOOLS

After a model is estimated, you can manipulate the results, compute predictions and marginal effects, or other functions of estimated parameters and model features. A built-in routine will compute nonlinear functions of your model estimates and associated standard errors with no programming by you other than specification of the desired function.

This section provides overall descriptions and lists of program features used in the different types of data analysis. Model estimation and hypothesis testing constitute the majority of this, but the other tools are frequently used as well.

MODEL ESTIMATION

TESTING AND RESTRICTIONS

Restricted estimation

·          Equality and fixed value restrictions in all models

·          Linearly constrained maximum likelihood

·          Inequality constrained least squares

Hypothesis tests

·          Wald test for linear restrictions

·          LM tests for any model using restricted starting values

·          LR tests for all models

·          Hausman test

o         Conditional logit

o         Linear panel data model

o         2SLS

o         IIA test for conditional logit

·        LM tests  

o       Tobit specification

o       Linear panel data models

·          Cox tests

·          PE and J tests

·        Vuong test for nonnested models

POST ESTIMATION ANALYSIS

PREDICTIONS

MARGINAL EFFECTS

DELTA METHODS

PROGRAMMING AND NUMERICAL ANALYSIS

A set of programming instructions including matrix and data manipulation commands and a nonlinear optimization program allows you to specify and estimate your own model if it is not included in the catalog of supported specifications.

This section describes tools that are used to create new estimators and to analyze results produced by the model estimation programs.

PROGRAMMING WITH LIMDEP

USER DEFINED OPTIMIZATION

MATRIX ALGEBRA

SCIENTIFIC CALCULATOR

USER WRITTEN PROGRAMS AND ESTIMATORS

NUMERICAL ANALYSIS TOOLS

Numerical analysis tools are provided for analyzing functions of a variable and functions of the data:

·          Function evaluation

·          Function plotting

·          Compute derivatives and save as gradient vector

·          Integration by quadrature

·          Integration by trapezoids

·          Integration by simulation

·          Matrix algebra

o         Characteristic roots

o         Roots of dynamic equations

o         Character vectors

o         Singular value decomposition

·          Scientific calculator: dozens of functions

·          Values of and asymptotic variances for nonlinear functions

CROSS SECTION, PANEL DATA, AND TIME SERIES FEATURES

Nearly every model supported by the program for cross section analysis may also be extended to a variety of panel data frameworks, including fixed and random effects, hierarchical (random parameters) and latent class formulations. There are also a variety of tools for time series analysis.

This section describes program features related to different types of data: cross section, panel data, and time series.

DYNAMIC LINEAR MODELS

This estimator in LIMDEP is built in two parts,

·          Hausman and Taylor instrumental variable estimator for the linear panel model

Hausman and Taylor’s estimator for the random effects model overcomes the possible correlation between the independent variables and the random effects. The random effects model is formulated with the possibility that there may be time invariant independent variables.

There are four sets of variables in the model, the xs which are time varying and the fs which are not time varying, and variables subscripted ‘1’ which are uncorrelated with ui and the remainder which may be correlated with ui.  A three step procedure which ends with generalized instrumental variables estimation is used for estimation.

·          Arellano/Bond/Bover estimator for the dynamic model.

The Arellano/Bond/Bover estimator is for the dynamic random effects model

This extends the Hausman and Taylor estimator. Two step GMM is used to estimate the model. You have a choice of covariance structures for ui (uncorrelated, random effect, freely correlated across time) and a choice of different sets of instrumental variables for the GMM estimator (variations in the number of future and lagged values). The optimal weighting matrix is computed at the second step. This is their efficient estimator. Barghava and Sargan’s specification test statistic is presented after estimation.

CONDITIONAL FIXED EFFECTS ESTIMATORS

A small number of fixed effects econometric models can be estimated by conditioning the fixed effects out of the model. These, which can all be fit with LIMDEP, are

·          Linear regression (use group mean deviations)

·          Binomial logit (condition on sum of outcomes)

·          Poisson (condition on sum of outcomes)

·          Negative binomial (conditional model is free of effects)

Note that Hausman et al’s negative binomial model is not a true fixed effects model in the fashion these are usually formulated, that is, in terms of an index function of the form a(i) + b’x(I,t). You can see this by trying to fit their model with its full set of dummy variables and an overall constant term - it works but it shouldn’t. The unconditional fixed effects model described in the section on the unconditional models is a true fixed effects model in that the effect acts to shift the mean.

LIMDEP computes conditional maximum likelihood estimates of all these models.

FIXED EFFECTS MODELS

We define fixed effects models in terms of the density of the observed random variable and an index function,

Density of observed y_it  =  f(y_it, a_i  +  b’x_it, other parameters)

There is one dummy variable coefficient for each individual or group. Familiar treatments in the literature are the linear regression model, in which the dummy variables are removed by deviations or by first differences, or in the binary logit or Poisson regression, in which the dummy variables are conditioned out and not estimated. LIMDEP’s implementation of this model is unconditional. All the dummy variable coefficients are actually estimated, with up to 20,000 groups, as well as the other model parameters. This is a new estimation method in Version 8.0 of LIMDEP that does not appear in any other software. This allows a far wider range of models than the conditional estimator:

·          Linear regression model

·          Probit, logit, Gompertz, complementary log log binary choice

·          Tobit, truncated regression, categorical data

·          Stochastic frontier

·          Survival models: exponential, Weibull, lognormal, loglogistic

·          Loglinear models: Weibull, gamma, exponential, inverse Gauss

·          Bivariate probit, partial observability

·          Ordered probit, ordered logit, ordered Gompertz, ordered complementary log log

·          Sample selection

·          Poisson, negative binomial, zero inflated Poisson

·          Conditional logit (multinomial logit - discrete choice)

Compare the preceding list to the list of conditional fixed effects estimators in LIMDEP and other programs: linear regression, binary logit, Poisson, negative binomial.

Other features of the unconditional fixed effects estimator include:

·          Full maximum likelihood estimation

·          Automatic data check for groups of one and groups with no variation

·          Dummy variable coefficients are retained

·          Predictions

·          Marginal effects

RANDOM EFFECTS MODELS

Random effects models are of the form:

       Density of observed dependent variable is f[y(I,t)|x(I,t),u(i)],

where x(I,t) is the vector of observed covariates and u(i) is a time invariant random variable. Random effects models are fit by two step generalized least squares for the linear model or by maximizing the unconditional likelihood for nonlinear models. The unconditional likelihood function is obtained by integrating u(i) out of the conditional likelihood. LIMDEP contains random effects estimators that use the familiar techniques in the literature for the following models.

·          Binomial probit, logit, Gompertz, complementary log log

·          Linear regression

·          Tobit, truncated regression, categorical data regression

·          Half- and truncated normal stochastic frontier

·          Ordered probit, logit, Gompertz, complementary log log

·          Poisson and negative binomial

Integration is done using closed forms, Gauss-Hermite quadrature (Butler-Moffitt) or maximum simulated likelihood: pseudo random draws or Halton sequences.

In addition to the preceding models, you can also fit a ‘random effects’ model by fitting a random parameters model which contains only a random constant term. This adds about 20 models to the list above.

·          Linear regression model

·          Probit, logit, Gompertz, complementary log log binary choice

·          Tobit, truncated regression, categorical data

·          Stochastic frontier

·          Survival models: exponential, Weibull, lognormal, loglogistic

·          Loglinear models: Weibull, gamma, exponential, inverse Gauss

·          Bivariate probit, partial observability

·          Ordered probit, ordered logit, ordered Gompertz, ordered complementary log log

·          Sample selection

·          Poisson, negative binomial, zero inflated Poisson

·          Conditional logit (multinomial logit – discrete choice)

RANDOM PARAMETERS – MULTILEVEL MODELS

The random parameters model is defined in terms of the density of the observed random variable and the structural parameters in the model:

Features05   

The random effects model is a special case in which only the constant term is random. The random parameters model has been implemented in other software for the binary logit, linear regression, and Poisson regression model. LIMDEP’s implementation supports a far wider variety of models.

·          Linear regression model

·          Probit, logit, Gompertz, complementary log log binary choice

·          Tobit, truncated regression, categorical data

·          Stochastic frontier

·          Survival models: exponential, Weibull, lognormal, loglogistic

·          Loglinear models: Weibull, gamma, exponential, inverse Gauss

·          Bivariate probit, partial observability

·          Ordered probit, ordered logit, ordered Gompertz, ordered complementary log log

·          Sample selection

·          Poisson, negative binomial, zero inflated Poisson

·          Conditional logit (multinomial logit - discrete choice)

Other features of the estimator:

·          Mixture of fixed and random parameters - you specify which parameters are random and which are fixed.

·          Panel data or cross section implementation

·          Distributions of random parameters may be normal, tent, uniform, lognormal

·          Maximum simulated likelihood may use pseudorandom draws or Halton sequences

·          ui may be a single random draw or AR(1)

·          Free correlation among random parameters (even with different distributions)

·          Predictions computed

·          Marginal effects

LATENT CLASS MODELS

The latent class model framework is of the form:

Model | class = familiar structure
Prob[class] = probability model

The class membership is not observed by the analyst. The estimated ‘model’ is therefore a weighted average of the underlying structures. LIMDEP’s implementation of this structure uses a multinomial logit model for the class probabilities, which allows observed covariates to influence the class probabilities. The models supported are:

·          Linear regression model

·          Probit, logit, Gompertz, complementary log log binary choice

·          Tobit, truncated regression, categorical data

·          Stochastic frontier

·          Survival models: exponential, Weibull, lognormal, loglogistic

·          Loglinear models: Weibull, gamma, exponential, inverse Gauss

·          Bivariate probit, partial observability

·          Ordered probit, ordered logit, ordered Gompertz, ordered complementary log log

·          Sample selection

·          Poisson, negative binomial, zero inflated Poisson

·          Conditional logit (multinomial logit - discrete choice)

Results produced include the J class specific parameter vectors for the model (you specify J), and the parameters of the probability model. The estimator is maximum likelihood. (The literature contains a few applications of this model to the linear regression, Poisson and binomial logit and probit models. The preceding extend the model into several new areas.) After estimation, you can compute an estimate of the class from which an observation is drawn (the one with the largest probability), and posterior estimates of the set of class probabilities. A suitably weighted average produces an individual (group) specific parameter vector as well. Results can also include a listing of the class probabilities. Post estimation results can also include predictions, marginal effects, etc.

TIME SERIES FEATURES

Model frameworks

·          Box-Jenkins ARIMA

·          ARMAX

·          ARCH, GARCH, GARCH(m)

·          Geometric lag model

Tools

·          Identification: ACF and PACF

·          Box-Pierce and Box-Ljung tests

·          Dates: yearly, quarterly, monthly

·          Unit root tests: Phillips and Perron

·          Dickey-Fuller test

·          Spectral density estimator

·          CUSUM test

Statistics

·          Autocorrelation

·          Partial autocorrelation

·          Phillips-Perron unit root tests

·          Spectral density estimator

Regression

·          ARMA and ARMAX regression

·          GARCH and GARCH in mean

·          Durbin-Watson test

·          Godfrey/Breusch-Pagan test

·          CUSUM and CUSUM squares tests

·          Autocorrelation: Prais-Winsten, maximum likelihood

·          AR(p) regression

·          Hatanaka estimator

Graphics

·          Line plots

·          Multiple time series plots

·          Plot forecasts from ARMAX models

·          CUSUM plot

·          Time series dating