olsmt¶

Purpose¶

Computes a least squares regression.

Format¶

out = olsmt(data, formula[, ctl])
out = olsmt(data, depvar, indvars[, ctl])
Parameters:
• data (string or dataframe) – name of dataset, dataframe in memory, or an empty string. If data is an empty string,””, the procedure assumes that the actual data has been passed in the next two arguments.

• formula (string) –

formula string of the model. E.g "y ~ X1 + X2", y is the name of dependent variable, X1 and X2 are names of independent variables;

E.g "y ~ .", . means including all variables except dependent variable y;

E.g "y ~ -1 + X1 + X2", -1 means no intercept model.

• depvar

If data contains a string, then depvar can be a:

type

value

string

name of dependent variable

scalar

index of dependent variable. If scalar 0, the last column of the dataset will be used.

If data is an empty string, name of dataframe, or 0:

type

value

Nx1 vector

the dependent variable.

• indvars (Kx1 vector or NxK matrix) –

If data contains a string:

type

value

Kx1 character vector

names of independent variables

Kx1 numeric vector

indices of independent variables. These can be any size subset of the variables in the data and can be in any order. If a scalar 0 is passed, all columns of the data will be used except for the one used for the dependent variable.

If data is an empty string, dataframe, or 0:

type

value

NxK matrix

the independent variables

• ctl (struct) –

Optional input. instance of an olsmtControl structure containing the following members:

 ctl.altnam string array, default "". This can be a $$(K+1) \times 1$$ or $$(K+2) \times 1$$ string array of alternate variable names for the output. If ctl.con is 1, and ctl.altnam has $$(K+2)$$ elements, then the first element will control the name displayed for the constant term. The name of the dependent variable is the last element. ctl.con scalar, default 1. 1: a constant term will be added, $$D = K+1$$. 0: no constant term will be added, $$D = K$$. A constant term will always be used in constructing the moment matrix m. ctl.cov string, set covariance type. Default = “iid”. "iid": Error terms assumed to be identical independently distributed. "robust": Huber/White/sandwich estimator. "cluster": Clustered sandwich estimator. Must specify cluster variable identifier. "hac": HAC robust sandwich estimator. ctl.clusterID Matrix, vector of categorical group variable used for computing cluster robust standard errors. ctl.clusterVar String, name of cluster group variable. Only valid if dataset and formula is specified. ctl.miss scalar, default 0. 0: there are no missing values (fastest). 1: listwise deletion, drop any cases in which missings occur. 2: pairwise deletion, this is equivalent to setting missings to 0 when calculating m. The number of cases computed is equal to the total number of cases in the dataset. ctl.row scalar, the number of rows to read per iteration of the read loop. Default 0. If 0, the number of rows will be calculated internally. If you get an Insufficient memory error message while executing olsmt(), you can supply a value for ctl.row that works on your system. The answers may vary slightly due to rounding error differences when a different number of rows is read per iteration. You can use ctl.row to control this if you want to get exactly the same rounding effects between several runs. ctl.vpad scalar, default 1. If 0, internally created variable names are not padded to the same length (e.g. X1, X2,..., X10). If 1, they are padded with zeros to the same length (e.g., X01, X02,..., X10). ctl.output scalar, default 1. 1: print the statistics. 0: do not print statistics. ctl.res scalar, default 0. 1: compute residuals (oOut.resid) and Durbin-Watson statistic (oOut.dwstat.) 0: oOut.resid = 0, oOut.dwstat = 0. ctl.rnam string, default “_olsmtres”. If the data is taken from a dataset, a new dataset will be created for the residuals, using the name in ctl.rnam. ctl.maxvec scalar, default 20000. The largest number of elements allowed in any one matrix. ctl.fcmptol scalar, default 1e-12. Tolerance used to fuzz the comparison operations to allow for round off error. ctl.alg string, default “cholup”. Selects the algorithm used for computing the parameter estimates. The default Cholesky update method is more computationally efficient. However, accuracy can suffer for poorly conditioned data. For higher accuracy set ctl.alg to either qr or svd. "qr": Solves for the parameter estimates using a qr decomposition. "svd": Solves for the parameter estimates using a singular value decomposition. ctl.weights Nx1 Vector, if defined, specifies weights to be used in the weighted least squares. If not defined, ordinary least squares will be computed. ctl.weightsVar String, name of the variable used for weighting. Only valid if dataset and formula is specified. Will override any weights in ctl.weights.

Returns:

out (struct) –

instance of olsmtOut struct containing the following members:

out.vnam

$$(K+2) \times 1$$ or $$(K+1) \times 1$$ character vector, the variable names used in the regression. If a constant term is used, this vector will be $$(K+2) \times 1$$, and the first name will be CONSTANT. The last name will be the name of the dependent variable.

out.m

MxM matrix, where $$M = K+2$$, the moment matrix constructed by calculating X'X where X is a matrix containing all useable observations and having columns in the order:

 1.0 indvars depvar (constant) (independent variables) (dependent variable)

A constant term is always used in computing m.

out.b

Dx1 vector, the least squares estimates of parameters.

Error handling is controlled by the low order bit of the trap flag.

trap 0:

terminate with error message

trap 1:

return scalar error code in b

 30 system singular 31 system underdetermined 32 same number of columns as rows 33 too many missings 34 file not found 35 no variance in an independent variable

The system can become underdetermined if you use listwise deletion and have missing values. In that case, it is possible to skip so many cases that there are fewer usable rows than columns in the dataset.

out.stb

Kx1 vector, the standardized coefficients.

out.vc

DxD matrix, the variance-covariance matrix of estimates.

out.stderr

Dx1 vector, the standard errors of the estimated parameters.

out.sigma

scalar, standard deviation of residual.

out.cx

$$(K+1) \times (K+1)$$ matrix, correlation matrix of variables with the dependent variable as the last column.

out.rsq

scalar, R square, coefficient of determination.

out.resid

residuals, $$out.resid = y - x * out.b$$.

If ctl.olsres = 1, the residuals will be computed.

If the data is taken from a dataset, a new dataset will be created for the residuals, using the name in ctl.rnam. The residuals will be saved in this dataset as an Nx1 column. The out.resid return value will be a string containing the name of the new dataset containing the residuals. If the data is passed in as a matrix, the out.resid return value will be the Nx1 vector of residuals.

out.dwstat

scalar, Durbin-Watson statistic.

Examples¶

Basic usage with matrices¶

// Set y matrix
y = { 2,
3,
1,
7,
5 };

//  Set x matrix
x = { 1 3 2,
2 3 1,
7 1 7,
5 3 1,
3 5 5 };

// Perform least squares regression and print report to the screen
// The empty string, "" indicates that no dataset is used
call olsmt("", y, x);


Basic usage with a dataset and a formula string¶

// Create string with the name and full file path of the dataset
dataset = getGAUSSHome("examples/detroit.dta");

// Create formula string specifying dependent and independent variables
formula  = "homicide ~ unemployment + hourly_earn";

// Perform estimation
call olsmt(dataset, formula);


In this example, the dataset detroit.dta is used to compute a regression. The dependent variable is homicide. The independent variables are: unemployment and hourly_earn. The output is:

Valid cases:                    13      Dependent variable:            homicide
Missing cases:                   0      Deletion method:                   None
Total SS:                 3221.790      Degrees of freedom:                  10
R-squared:                   0.834      Rbar-squared:                     0.801
Residual SS:               533.814      Std error of est:                 7.306
F(2,10):                    25.177      Probability of F:                 0.000

Standard                 Prob   Standardized  Cor with
Variable     Estimate      Error      t-value     >|t|     Estimate    Dep Var
-----------------------------------------------------------------------------------

CONSTANT       -35.982790    9.437246   -3.812849     0.003       ---         ---
unemployment    -0.004998    0.918817   -0.005440     0.996   -0.000720    0.210142
hourly_earn     15.487191    2.242660    6.905722     0.000    0.913572    0.913406


Basic usage with a dataframe and categorical variable¶

// Load data
fname = getGAUSSHome("examples/auto2.dta");

// Include the rep78 categorical variable
call olsmt(auto2, "price ~ mpg + rep78");


In this example, the dependent variable price is regressed on mpg and rep78. The categorical variable rep78 will automatically be included in the OLS regression as a dummy variable with the base case excluded from the regression. The coefficients for the categories, Fair, Average, Good, Excellent are included in the printed output table. The Poor category is excluded from the regression, as it is the base case.

Valid cases:                    69      Dependent variable:               price
Missing cases:                   5      Deletion method:               Listwise
Total SS:            576796958.870      Degrees of freedom:                  63
R-squared:                   0.258      Rbar-squared:                     0.199
Residual SS:         427776355.434      Std error of est:              2605.782
F(5,63):                     4.389      Probability of F:                 0.002

Standard                 Prob   Standardized  Cor with
Variable             Estimate      Error      t-value     >|t|     Estimate    Dep Var
---------------------------------------------------------------------------------------

CONSTANT                10450     2251.04     4.64229     0.000       ---         ---
mpg                  -280.261     61.5767    -4.55142     0.000   -0.564519   -0.455949
rep78: Fair           877.635     2063.28    0.425358     0.672   0.0971824  -0.0223477
rep78: Average        1425.66     1905.44    0.748204     0.457     0.24444   0.0859051
rep78: Good           1693.84     1942.67    0.871914     0.387    0.257252   -0.015317
rep78: Excellent      3131.98     2041.05      1.5345     0.130    0.396546   -0.035102


Estimate a linear model for each subset of a categorical variable¶

In this example, we will regress mpg and weight on price for the case where foreign equals “Domestic” and for the case where foreign equals “Foreign”, using the by keyword.

// Load specific variables
fname = getGAUSSHome("examples/auto2.dta");
auto2 = loadd(fname, "price + mpg + weight + foreign");

// Estimate a linear model for each value of foreign
call olsmt(auto2, "price ~ mpg + weight + by(foreign)");

===============================================================================
foreign: Domestic
===============================================================================
Valid cases:                    52      Dependent variable:               price
Missing cases:                   0      Deletion method:                   None
Total SS:            489194800.692      Degrees of freedom:                  49
R-squared:                   0.483      Rbar-squared:                     0.462
Residual SS:         252934086.227      Std error of est:              2271.986
F(2,49):                    22.885      Probability of F:                 0.000

Standard                 Prob   Standardized  Cor with
Variable     Estimate      Error      t-value     >|t|     Estimate    Dep Var
-------------------------------------------------------------------------------

CONSTANT     -13285.4     5726.03    -2.32018     0.025       ---         ---
mpg           237.691     139.033      1.7096     0.094     0.36403   -0.504263
weight        4.41504    0.948391     4.65529     0.000    0.991267    0.672397

===============================================================================
foreign: Foreign
===============================================================================
Valid cases:                    22      Dependent variable:               price
Missing cases:                   0      Deletion method:                   None
Total SS:            144363212.773      Degrees of freedom:                  19
R-squared:                   0.785      Rbar-squared:                     0.763
Residual SS:          30967505.235      Std error of est:              1276.663
F(2,19):                    34.787      Probability of F:                 0.000

Standard                 Prob   Standardized  Cor with
Variable     Estimate      Error      t-value     >|t|     Estimate    Dep Var
-------------------------------------------------------------------------------

CONSTANT     -5065.84     3202.51    -1.58183     0.130       ---         ---
mpg          -19.7774     57.6812   -0.342874     0.735  -0.0498688   -0.631303
weight        5.15584    0.880689     5.85433     0.000    0.851476    0.885529


Use a dataset, a list of variable names plus a control and output structure.¶

new;

// Declare 'ols_ctl' to be an olsmtControl structure
// and fill with default settings
struct olsmtControl ols_ctl;
ols_ctl = olsmtControlCreate();

// Set the 'res' member of the olsmtControl structure
// so that 'olsmt' will compute residuals and the Durbin-Watson statistic
ols_ctl.res = 1;

// Declare 'ols_out' to be an olsmtOut structure
// to hold the results of the computations
struct olsmtOut ols_out;

// Create string with the name and full file path of the dataset
data = getGAUSSHome("examples/credit.dat");

// Create a string with the name of the dependent variable
depvar = "Limit";

// Create 3x1 string array, containing the dependent variable names
indvars = "Balance" $| "Income"$|  "Age";

// Perform estimation, using settings in the 'ols_ctl'
// control structure and store the results in 'ols_out'
ols_out = olsmt(data, depvar, indvars, ols_ctl);


In this example, the dataset credit.dat is used to compute a regression. The dependent variable is Limit. The independent variables are: Balance, Income, and Age. The residuals and Durbin-Watson statistic will be computed.

Use a dataset and variable indices¶

// Set dataset name
dataset = getGAUSSHome("examples/credit.dat");

// Set the third variable in 'credit.dat', 'Rating'
// to be the dependent variable
depvar = 3;

// Set the first, second and fifth variables in 'credit.dat'
// to be the independent variables
indepvar = { 1, 2, 5 };

call olsmt(dataset, depvar, indepvar);


The above code will produce the following output:

Valid cases:                   400      Dependent variable:              Rating
Missing cases:                   0      Deletion method:                   None
Total SS:              9551884.560      Degrees of freedom:                 396
R-squared:                   0.994      Rbar-squared:                     0.994
Residual SS:             59390.952      Std error of est:                12.247
F(3,396):                21097.644      Probability of F:                 0.000

Standard                 Prob   Standardized  Cor with
Variable     Estimate      Error      t-value     >|t|     Estimate    Dep Var
-------------------------------------------------------------------------------
CONSTANT    37.675546    2.415716   15.596014     0.000       ---         ---
Income       0.018253    0.028857    0.632538     0.527    0.004158    0.791378
Limit        0.066587    0.000436  152.717620     0.000    0.993363    0.996880
Age          0.019892    0.036174    0.549896     0.583    0.002218    0.103165


Basic usage with weights¶

new;

// Define data
parent = { 0.21, 0.2, 0.19, 0.18, 0.17, 0.16, 0.15 };
progeny = { 0.1726, 0.1707, 0.1637, 0.164, 0.1613, 0.1617, 0.1598 };
sd = { 0.01988, 0.01938, 0.01896, 0.02037, 0.01654, 0.01594, 0.01763 };

// Calculate weights
weights = 1 ./ SD.^2;

// Set up olsControl structure
struct olsmtControl ctl;
ctl = olsmtControlCreate();
ctl.weights = weights;

call olsmt("", progeny, parent, ctl);


The above code will produce the following output:

Valid cases:                     7      Dependent variable:                   Y
Missing cases:                   0      Deletion method:                   None
Total SS:                  572.494      Degrees of freedom:                   5
R-squared:                   0.852      Rbar-squared:                     0.823
Residual SS:                 0.061      Std error of est:                 0.110
F(1,5):                     28.812      Probability of F:                 0.002

Standard                 Prob   Standardized  Cor with
Variable     Estimate      Error      t-value     >|t|     Estimate    Dep Var
-------------------------------------------------------------------------------

CONSTANT     0.127964  0.00681124     18.7872     0.000    0.778572    0.999643
X1           0.204801   0.0381548     5.36763     0.003    0.222444    0.996209


Remarks¶

• For poorly conditioned data the default setting for ctl.olsalg, using the Cholesky update, may produce only four or five digits of accuracy for the parameter estimates and standard error. For greater accuracy, use either the qr or singular value decomposition algorithm by setting ctl.olsalg to qr or svd. If you are unsure of the condition of your data, set ctl.olsalg to qr.

• No output file is modified, opened, or closed by this procedure. If you want output to be placed in a file, you need to open an output file before calling olsmt().

• The supported dataset types are CSV, XLS, XLSX, HDF5, FMT, DAT

• For HDF5 file, the dataset must include file schema and both file name and dataset name must be provided, e.g.

olsmt("h5://C:/gauss/examples/testdata.h5/mydata", formula)


olsmt.src