fgls
Feasible generalized least squares
Syntax
Description
[
returns vectors of coefficient estimates and corresponding standard errors, and the
estimated coefficient covariance matrix, from applying feasible generalized least squares (FGLS) to the multiple linear regression
model coeff
,se
,EstCoeffCov
] = fgls(X
,y
)y
= X
β +
ε. y
is a vector of response data and
X
is a matrix of predictor data.
[
returns a table containing FGLS coefficient estimates and corresponding standard errors,
and a table containing the FGLS estimated coefficient covariance matrix, from applying
FGLS to the variables in the input table or timetable. The response variable in the
regression is the last variable, and all other variables are the predictor
variables.CoeffTbl
,CovTbl
] = fgls(Tbl
)
To select a different response variable for the regression, use the
ResponseVariable
name-value argument. To select different predictor
variables, use the PredictorNames
name-value argument.
[___] = fgls(___,
specifies options using one or more name-value arguments in
addition to any of the input argument combinations in previous syntaxes.
Name=Value
)fgls
returns the output argument combination for the
corresponding input arguments.
For example,
fgls(Tbl,ResponseVariable="GDP",InnovMdl="H4",Plot="all")
provides
coefficient, standard error, and residual mean-squared error (MSE) plots of iterations of
FGLS for a regression model with White’s robust innovations covariance, and the table
variable GDP
is the response while all other variables are
predictors.
[___] = fgls(
plots on the axes specified in ax
,___,Plot=plot)ax
instead of the axes of new figures
when plot
is not "off"
. ax
can
precede any of the input argument combinations in the previous syntaxes.
[___,
returns handles to plotted graphics objects when iterPlots
] = fgls(___,Plot=plot)plot
is not
"off"
. Use elements of iterPlots
to modify
properties of the plots after you create them.
Examples
Estimate FGLS Coefficients and Uncertainty Measures
Suppose the sensitivity of the US consumer price index (CPI) to changes in the paid compensation of employees (COE) is of interest.
Load the US macroeconomic data set, which contains the timetable of data DataTimeTable
. Extract the COE and CPI series from the table.
load Data_USEconModel.mat
COE = DataTimeTable.COE;
CPI = DataTimeTable.CPIAUCSL;
dt = DataTimeTable.Time;
Plot the series.
tiledlayout(2,1) nexttile plot(dt,CPI); title("\bf Consumer Price Index, Q1 in 1947 to Q1 in 2009"); axis tight nexttile plot(dt,COE); title("\bf Compensation Paid to Employees, Q1 in 1947 to Q1 in 2009"); axis tight
The series are nonstationary. Stabilize them by computing their returns.
rCPI = price2ret(CPI); rCOE = price2ret(COE);
Regress rCPI
onto rCOE
including an intercept to obtain ordinary least squares (OLS) estimates, standard errors, and the estimated coefficient covariance. Generate a lagged residual plot.
Mdl = fitlm(rCOE,rCPI); clmCoeff = Mdl.Coefficients.Estimate
clmCoeff = 2×1
0.0033
0.3513
clmSE = Mdl.Coefficients.SE
clmSE = 2×1
0.0010
0.0490
CLMEstCoeffCov = Mdl.CoefficientCovariance
CLMEstCoeffCov = 2×2
0.0000 -0.0000
-0.0000 0.0024
figure
plotResiduals(Mdl,"lagged")
The residual plot exhibits an upward trend, which suggests that the innovations comprise an autoregressive process. This violates one of the classical linear model assumptions. Consequently, hypothesis tests based on the regression coefficients are incorrect, even asymptotically.
Estimate the regression coefficients, standard errors, and coefficient covariances using FGLS. By default, fgls
includes an intercept in the regression model and imposes an AR(1) model on the innovations.
[coeff,se,EstCoeffCov] = fgls(rCPI,rCOE)
coeff = 2×1
0.0148
0.1961
se = 2×1
0.0012
0.0685
EstCoeffCov = 2×2
0.0000 -0.0000
-0.0000 0.0047
Row 1 of the outputs corresponds to the intercept and row 2 corresponds to the coefficient of rCOE
.
If the COE series is exogenous with respect to the CPI, then the FGLS estimates coeff
are consistent and asymptotically more efficient than the OLS estimates.
Estimate FGLS Coefficients and Uncertainty Measures on Table Data
Load the US macroeconomic data set, which contains the timetable of data DataTimeTable
.
load Data_USEconModel
Stabilize all series by computing their returns.
RDT = price2ret(DataTimeTable);
RDT
is a timetable of returns of all variables in DataTimeTable
. The price2ret
function conserves variable names.
Estimate the regression coefficients, standard errors, and the coefficient covariance matrix using FGLS. Specify the response and predictor variable names.
[CoeffTbl,CoeffCovTbl] = fgls(RDT,ResponseVariable="CPIAUCSL",PredictorVariables="COE")
CoeffTbl=2×2 table
Coeff SE
__________ _________
Const 6.2416e-05 1.336e-05
COE 0.20562 0.055615
CoeffCovTbl=2×2 table
Const COE
___________ ___________
Const 1.7848e-10 -5.6329e-07
COE -5.6329e-07 0.003093
When you supply a table or timetable of data, fgls
returns tables of estimates.
Specify AR Lags For FGLS Estimation
Suppose the sensitivity of the US consumer price index (CPI) to changes in the paid compensation of employees (COE) is of interest. This example expands on the analysis outlined in the example Estimate FGLS Coefficients and Uncertainty Measures.
Load the US macroeconomic data set.
load Data_USEconModel
The series are nonstationary. Stabilize them by applying the log, and then the first difference.
LDT = price2ret(Data); rCOE = LDT(:,1); rCPI = LDT(:,2);
Regress rCPI
onto rCOE
, which includes an intercept to obtain OLS estimates. Plot correlograms for the residuals.
Mdl = fitlm(rCOE,rCPI); u = Mdl.Residuals.Raw; figure; subplot(2,1,1) autocorr(u); subplot(2,1,2); parcorr(u);
The correlograms suggest that the innovations have significant AR effects. According to Box-Jenkins methodology, the innovations seem to comprise an AR(3) series. For details, see Select ARIMA Model for Time Series Using Box-Jenkins Methodology.
Estimate the regression coefficients using FGLS. By default, fgls
assumes that the innovations are autoregressive. Specify that the innovations are AR(3) by using the ARLags
name-value argument, and print the final estimates to the command window by using the Display
name-value argument.
fgls(rCPI,rCOE,ARLags=3,Display="final");
OLS Estimates: | Coeff SE ------------------------ Const | 0.0122 0.0009 x1 | 0.4915 0.0686 FGLS Estimates: | Coeff SE ------------------------ Const | 0.0148 0.0012 x1 | 0.1972 0.0684
If the COE rate series is exogenous with respect to the CPI rate, the FGLS estimates are consistent and asymptotically more efficient than the OLS estimates.
Account for Residual Heteroscedasticity Using FGLS Estimation
Model the nominal GNP GNPN
growth rate accounting for the effects of the growth rates of the consumer price index CPI
, real wages WR
, and the money stock MS
. Account for classical linear model departures.
Load the Nelson-Plosser data set, which contains the data in the table DataTable
. Remove all observations containing at least one missing value.
load Data_NelsonPlosser
DT = rmmissing(DataTable);
T = height(DT);
Plot the series.
predNames = ["CPI" "WR" "MS"]; tiledlayout(2,2) for j = ["GNPN" predNames] nexttile plot(DT{:,j}); xticklabels(DT.Dates) title(j); axis tight end
All series appear nonstationary.
For each series, compute the returns.
RetDT = price2ret(DT);
RetTT
is a timetable of the returns of the variables in TT
. The variables names are conserved.
Regress the GNPN
rate onto the CPI
, WR
, and MS
rates. Examine a scatter plot and correlograms of the residuals.
Mdl = fitlm(RetDT,ResponseVar="GNPN",PredictorVar=predNames); figure plotResiduals(Mdl,"caseorder"); axis tight
figure tiledlayout(2,1) nexttile autocorr(Mdl.Residuals.Raw); nexttile parcorr(Mdl.Residuals.Raw);
The residuals appear to flare in, which is indicative of heteroscedasticity. The correlograms suggest that there is no autocorrelation.
Estimate FGLS coefficients by accounting for the heteroscedasticity of the residuals. Specify that the estimated innovation covariance is diagonal with the squared residuals as weights (that is, White's robust estimator H0
).
fgls(RetDT,ResponseVariable="GNPN",PredictorVariables=predNames, ... InnovMdl="HC0",Display="final");
OLS Estimates: | Coeff SE ------------------------- Const | -0.0076 0.0085 CPI | 0.9037 0.1544 WR | 0.9036 0.1906 MS | 0.4285 0.1379 FGLS Estimates: | Coeff SE ------------------------- Const | -0.0102 0.0017 CPI | 0.8853 0.0169 WR | 0.8897 0.0294 MS | 0.4874 0.0291
Estimate FGLS Coefficients of Models Containing ARMA Errors
Create this regression model with ARMA(1,2) errors, where is Gaussian with mean 0 and variance 1.
beta = [2 3];
phi = 0.2;
theta = [-0.3 0.1];
Mdl = regARIMA(AR=phi,MA=theta,Intercept=1, ...
Beta=beta,Variance=1);
Mdl
is a regARIMA
model. You can access its properties using dot notation.
Simulate 500 periods of 2-D standard Gaussian values for , and then simulate responses using Mdl
.
numObs = 500;
rng(1); % For reproducibility
X = randn(numObs,2);
y = simulate(Mdl,numObs,X=X);
fgls
supports AR(p) innovation models. You can convert an ARMA model polynomial to an infinite-lag AR model polynomial using arma2ar
. By default, arma2ar
returns the coefficients for the first 10 terms. After the conversion, determine how many lags of the resulting AR model are practically significant by checking the length of the returned vector of coefficients. Choose the number of terms that exceed 0.00001.
format long
arParams = arma2ar(phi,theta)
arParams = 1×3
-0.100000000000000 0.070000000000000 0.031000000000000
arLags = sum(abs(arParams) > 0.00001);
format short
Some of the parameters have small magnitude. You might want to reduce the number of lags to include in the innovations model for fgls
.
Estimate the coefficients and their standard errors using FGLS and the simulated data. Specify that the innovations comprise an AR(arLags
) process.
[coeff,~,EstCoeffCov] = fgls(X,y,InnovMdl="AR",ARLags=arLags)
coeff = 3×1
1.0372
2.0366
2.9918
EstCoeffCov = 3×3
0.0026 -0.0000 0.0001
-0.0000 0.0022 0.0000
0.0001 0.0000 0.0024
The estimated coefficients are close to their true values.
Plot Iterations of FGLS Estimation
This example expands on the analysis in Estimate FGLS Coefficients of Models Containing ARMA Errors. Create this regression model with ARMA(1,4) errors, where is Gaussian with mean 0 and variance 1.
beta = [1.5 2]; phi = 0.9; theta = [-0.4 0.2]; Mdl = regARIMA(AR=phi,MA=theta,MALags=[1 4],Intercept=1,Beta=beta,Variance=1);
Suppose the distribution of the predictors is
Simulate 30 periods from , and then simulate 30 corresponding responses from the regression model with ARMA errors Mdl
.
numObs = 30;
rng(1); % For reproducibility
muX = [-1 1];
sigX = [0.5 1];
X = randn(numObs,numel(beta)).*sigX + muX;
y = simulate(Mdl,numObs,X=X);
Convert the ARMA model polynomial to an infinite-lag AR model polynomial using arma2ar
. By default, arma2ar
returns the coefficients for the first 10 terms. Find the number of terms that exceed 0.00001.
arParams = arma2ar(phi,theta); arLags = sum(abs(arParams) > 1e-5);
Estimate the regression coefficients by using eight iterations of FGLS, and specify the number of lags in the AR innovation model (arLags
). Also, specify to plot the coefficient estimates and their standard errors for each iteration, and to display the final estimates and the OLS estimates in tabular form.
[coeff,~,EstCoeffCov] = fgls(X,y,InnovMdl="AR",ARLags=arLags, ... NumIter=8,Plot=["coeff" "se"],Display="final");
OLS Estimates: | Coeff SE ------------------------ Const | 1.7619 0.4514 x1 | 1.9637 0.3480 x2 | 1.7242 0.2152 FGLS Estimates: | Coeff SE ------------------------ Const | 1.0845 0.6972 x1 | 1.7020 0.2919 x2 | 2.0825 0.1603
The algorithm seems to converge after the four iterations. The FGLS estimates are closer to the true values than the OLS estimates.
Properties of iterative FGLS estimates in finite samples are difficult to establish. For asymptotic properties, one iteration of FGLS is sufficient, but fgls
supports iterative FGLS for experimentation.
If the estimates or standard errors show instability after successive iterations, then the estimated innovations covariance might be ill conditioned. Consider scaling the residuals by using the ResCond
name-value argument to improve the conditioning of the estimated innovations covariance.
Input Arguments
X
— Predictor data X
numeric matrix
Predictor data X for the multiple linear regression model,
specified as a numObs
-by-numPreds
numeric
matrix.
Each row represents one of the numObs
observations and each column
represents one of the numPreds
predictor variables.
Data Types: double
y
— Response data y
numeric vector
Response data y for the multiple linear
regression model, specified as a
numObs
-by-1 numeric vector.
Rows of y
and
X
correspond.
Data Types: double
Tbl
— Combined predictor and response data
table | timetable
Combined predictor and response data for the multiple linear regression model,
specified as a table or timetable with numObs
rows. Each row of
Tbl
is an observation.
The test regresses the response variable, which is the last variable in
Tbl
, on the predictor variables, which are all other variables
in Tbl
. To select a different response variable for the regression,
use the ResponseVariable
name-value argument. To select different
predictor variables, use the PredictorNames
name-value argument to
select numPreds
predictors.
ax
— Axes on which to plot
vector of Axes
objects
Axes on which to plot, specified as a vector of Axes
objects with
length equal to the number of plots specified by the Plot
name-value argument.
By default, fgls
creates a separate figure for each
plot.
Note
NaN
s in X
,
y
, or Tbl
indicate missing values, and
fgls
removes observations containing at least one
NaN
. That is, to remove NaN
s in X
or y
, fgls
merges the variables [X
y]
, and then it uses list-wise deletion to remove any row that contains at least
one NaN
. fgls
also removes any row of
Tbl
containing at least one NaN
. Removing
NaN
s in the data reduces the sample size and can create irregular time
series.
Name-Value Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
Example: fgls(Tbl,ResponseVariable="GDP",InnovMdl="H4",Plot="all")
provides coefficient, standard error, and residual mean squared error (RMSE) plots
of iterations of FGLS for a regression model with White’s robust innovations
covariance, and the table variable GDP
is the response while all
other variables are predictors.
VarNames
— Unique variable names to use in display
string vector | character vector | cell vector of strings | cell vector of character vectors
Unique variable names used in the display, specified as a string vector or cell
vector of strings of a length numCoeffs
:
If
Intercept=true
,VarNames(1)
is the name of the intercept (for example'Const'
) andVarNames(
specifies the name to use for variablej
+ 1)X(:,
orj
)PredictorVariables(
.j
)If
Intercept=false
,VarNames(
specifies the name to use for variablej
)X(:,
orj
)PredictorVariables(
.j
)
The defaults is one of the following alternatives prepended by
'Const'
when an intercept is present in the model:
Example: VarNames=["Const" "AGE" "BBD"]
Data Types: char
| cell
| string
Intercept
— Flag to include intercept
true
(default) | false
Flag to include a model intercept, specified as a value in this table.
Value | Description |
---|---|
true | fgls includes an intercept term in the regression model. numCoeffs = numPreds + 1. |
false | fgls does not include an intercept when fitting the regression model. numCoeffs = numPreds . |
Example: Intercept=false
Data Types: logical
InnovMdl
— Model for innovations covariance estimate
"AR"
(default) | "CLM"
| "HC0"
| "HC1"
| "HC2"
| "HC3"
| "HC4"
| character vector
Model for the innovations covariance estimate, specified as a model name in the following table.
Set InnovMdl
to specify the structure of the
innovations covariance estimator .
For diagonal innovations covariance models (i.e., models with heteroscedasticity), where ω = {ωi; i = 1,...,T} is a vector of innovation variance estimates for the observations, and T =
numObs
.fgls
estimates the data-driven vector ω using the corresponding model residuals (ε), their leverages and the degrees of freedom dfe.For full innovation covariance models (in other words, models having heteroscedasticity and autocorrelation), specify
"AR"
.fgls
imposes an AR(p) model on the innovations, and constructs using the number of lags, p, specified by the name-value argumentarLags
and the Yule-Walker equations.
If the NumIter
name-value argument is
1
and you specify the
InnovCov0
name-value argument,
fgls
ignores
InnovMdl
.
Example: InnovMdl=HC0
Data Types: char
| string
ARLags
— Number of lags
1
(default) | positive integer
Number of lags to include in the autoregressive (AR) innovations model, specified as a positive integer.
If the InnovMdl
name-value argument is not
"AR"
(that is, for diagonal models),
fgls
ignores
ARLags
.
For general ARMA innovations models, convert the innovations model to the equivalent AR form by performing one of the following actions.
Construct the ARMA innovations model lag operator polynomial using
LagOp
. Then, divide the AR polynomial by the MA polynomial using, for example,mrdivide
. The result is the infinite-order, AR representation of the ARMA model.Use
arma2ar
, which returns the coefficients of the infinite-order, AR representation of the ARMA model.
Example: ARLags=4
Data Types: double
InnovCov0
— Initial innovations covariance
[]
(default) | positive vector | positive definite matrix | positive semidefinite matrix
Initial innovations covariance, specified as a positive vector, positive semidefinite matrix, or a positive definite matrix.
InnovCov0
replaces the data-driven estimate of the innovations covariance () in the first iteration of GLS.
For diagonal innovations covariance models (that is, models with heteroscedasticity), specify a
numObs
-by-1 vector.InnovCov0(
is the variance of innovationj
)j
.For full innovation covariance models (that is, models having heteroscedasticity and autocorrelation), specify a
numObs
-by-numObs
matrix.InnovCov0(
is the covariance of innovationsj
,k
)j
andk
.
By default, fgls
uses a data-driven (see the InnovMdl
name-value
argument).
Data Types: double
NumIter
— Number of iterations
1
(default) | positive integer
Number of iterations to implement for the FGLS algorithm, specified as a positive integer.
fgls
estimates the innovations covariance at each iteration from the residual series according to the
innovations covariance model InnovMdl
. Then, the software
computes the GLS estimates of the model coefficients.
Example: NumIter=10
Data Types: double
ResCond
— Flag to scale residuals
false
(default) | true
Flag to scale the residuals at each iteration of FGLS, specified as a value in this table.
Value | Description |
---|---|
true | fgls scales the residuals
at each iteration. |
false | fgls does not scale the
residuals at each iteration. |
Tip
The setting ResCond=true
can improve the
conditioning of the estimation of the innovations covariance .
Data Types: logical
Display
— Command window display control
"off"
(default) | "final"
| "iter"
| character vector
Command window display control, specified as a value in this table.
Value | Description |
---|---|
"final" | fgls displays the
final estimates. |
"iter" | fgls displays the
estimates after each iteration. |
"off" | fgls suppresses
command window display. |
fgls
shows estimation
results in tabular form.
Example: Display="iter"
Data Types: char
| string
Plot
— Control for plotting results
"off"
(default) | "all"
| "coeff"
| "mse"
| "se"
| character vector | string vector | cell array of character vectors
Control for plotting results after each iteration, specified as a value in the following table, or a string vector or cell array of character vectors of such values.
To examine the convergence of the FGLS algorithm, specify plotting the estimates for each iteration.
Value | Description |
---|---|
"all" | fgls plots the
estimated coefficients, their standard errors, and the residual mean-squared
error (MSE) on separate plots. |
"coeff" | fgls plots the
estimated coefficients. |
"mse" | fgls plots the
MSEs. |
"off" | fgls does not plot
the results. |
"se" | fgls plots the
estimated coefficient standard errors. |
Example: Plot="all"
Example: Plot=["coeff" "se"]
separately plots
iterative coefficient estimates and their standard
errors.
Data Types: char
| string
| cell
ResponseVariable
— Variable in Tbl
to use for response
first variable in Tbl
(default) | string vector | cell vector of character vectors | vector of integers | logical vector
Variable in Tbl
to use for response, specified as a string vector or cell vector of character vectors containing variable names in Tbl.Properties.VariableNames
, or an integer or logical vector representing the indices of names. The selected variables must be numeric.
fgls
uses the same specified response variable for all tests.
Example: ResponseVariable="GDP"
Example: ResponseVariable=[true false false false]
or
ResponseVariable=1
selects the first table variable as the
response.
Data Types: double
| logical
| char
| cell
| string
PredictorVariables
— Variables in Tbl
to use for the predictors
string vector | cell vector of character vectors | vector of integers | logical vector
Variables in Tbl
to use for the predictors, specified as a string vector or cell vector of character vectors containing variable names in Tbl.Properties.VariableNames
, or an integer or logical vector representing the indices of names. The selected variables must be numeric.
fgls
uses the same specified predictors for all tests.
By default, fgls
uses all variables in Tbl
that are not specified by the ResponseVariable
name-value
argument.
Example: PredictorVariables=["UN" "CPI"]
Example: PredictorVariables=[false true true false]
or
DataVariables=[2 3]
selects the second and third table
variables.
Data Types: double
| logical
| char
| cell
| string
Output Arguments
coeff
— FGLS coefficient estimates
numeric vector
FGLS coefficient estimates, returned as a numCoeffs
-by-1 numeric
vector. fgls
returns coeff
when you
supply the inputs X
and y
.
Rows of coeff
correspond to the predictor matrix columns, with
the first row corresponding to the intercept when Intercept=true
.
For example, in a model with an intercept, the value of (corresponding to the predictor
x1) is in position 2 of
coeff
.
se
— Coefficient standard error estimates
numeric vector
Coefficient standard error estimates, returned as a
numCoeffs
-by-1 numeric. The elements of se
are
sqrt(diag(EstCoeffCov))
. fgls
returns
se
when you supply the inputs X
and
y
.
Rows of se
correspond to the predictor matrix columns, with the
first row corresponding to the intercept when Intercept=true
. For
example, in a model with an intercept, the estimated standard error of (corresponding to the predictor
x1) is in position 2 of
se
, and is the square root of the value in position (2,2) of
EstCoeffCov
.
EstCoeffCov
— Coefficient covariance matrix estimate
numeric matrix
Coefficient covariance matrix estimate, returned as a
numCoeffs
-by-numCoeffs
numeric matrix.
fgls
returns EstCoeffCov
when you
supply the inputs X
and y
.
Rows and columns of EstCoeffCov
correspond to the predictor
matrix columns, with the first row and column corresponding to the intercept when
Intercept=true
. For example, in a model with an intercept, the
estimated covariance of (corresponding to the predictor
x1) and (corresponding to the predictor
x2) are in positions (2,3) and (3,2) of
EstCoeffCov
, respectively.
CoeffTbl
— FGLS coefficient estimates and standard errors
table
FGLS coefficient estimates and standard errors, returned as a
numCoeffs
-by-2 table. fgls
returns
CoeffTbl
when you supply the input
Tbl
.
For j
= 1,…,numCoeffs
, row
j
of CoeffTbl
contains estimates of
coefficient j
in the regression model and it has label
VarNames(
. The first variable
j
)Coeff
contains the coefficient estimates coeff
and the second variable SE
contains the standard errors
se
.
CovTbl
— Coefficient covariance matrix estimate
table
Coefficient covariance matrix estimate, returned as a
numCoeffs
-by-numCoeffs
table containing the
coefficient covariance matrix estimate EstCoeffCov
.
fgls
returns CovTbl
when you supply
the input Tbl
.
For each pair (i
,j
),
CovTbl(
contains the covariance estimate of coefficients i
,j
)i
and
j
in the regression model. The label of row and variable
j
is
VarNames(
,
j
)j
= 1,…,numCoeffs
.
iterPlots
— Handles to plotted graphics objects
structure array of graphics objects
Handles to plotted graphics objects, returned as a structure array of graphics
objects. iterPlots
contains unique plot identifiers, which you can
use to query or modify properties of the plot.
iterPlots
is not available if the value of the
Plot
name-value argument is "off"
.
More About
Feasible Generalized Least Squares
Feasible generalized least squares (FGLS) estimates the coefficients of a multiple linear regression model and their covariance matrix in the presence of nonspherical innovations with an unknown covariance matrix.
Let yt = Xtβ + εt be a multiple linear regression model, where the innovations process εt is Gaussian with mean 0, but with true, nonspherical covariance matrix Ω (for example, the innovations are heteroscedastic or autocorrelated). Also, suppose that the sample size is T and there are p predictors (including an intercept). Then, the FGLS estimator of β is
where is an innovations covariance estimate based on a model (e.g., innovations process forms an AR(1) model). The estimated coefficient covariance matrix is
where
FGLS estimates are computed as follows:
OLS is applied to the data, and then residuals are computed.
is estimated based on a model for the innovations covariance.
is estimated, along with its covariance matrix
Optional: This process can be iterated by performing the following steps until converges.
Compute the residuals of the fitted model using the FGLS estimates.
Apply steps 2–3.
If is a consistent estimator of and the predictors that comprise X are exogenous, then FGLS estimators are consistent and efficient.
Asymptotic distributions of FGLS estimators are unchanged by repeated iteration. However, iterations might change finite sample distributions.
Generalized Least Squares
Generalized least squares (GLS) estimates the coefficients of a multiple linear regression model and their covariance matrix in the presence of nonspherical innovations with known covariance matrix.
The setup and process for obtaining GLS estimates is the same as in FGLS, but replace with the known innovations covariance matrix .
In the presence of nonspherical innovations, and with known innovations covariance, GLS estimators are unbiased, efficient, and consistent, and hypothesis tests based on the estimates are valid.
Weighted Least Squares
Weighted least squares (WLS) estimates the coefficients of a multiple linear regression model and their covariance matrix in the presence of uncorrelated but heteroscedastic innovations with known, diagonal covariance matrix.
The setup and process to obtain WLS estimates is the same as in FGLS, but replace with the known, diagonal matrix of weights. Typically, the diagonal elements are the inverse of the variances of the innovations.
In the presence of heteroscedastic innovations, and when the variances of the innovations are known, WLS estimators are unbiased, efficient, and consistent, and hypothesis tests based on the estimates are valid.
Tips
To obtain standard generalized least squares (GLS) estimates:
To obtain weighted least squares (WLS) estimates, set the
InnovCov0
name-value argument to a vector of inverse weights (e.g., innovations variance estimates).In specific models and with repeated iterations, scale differences in the residuals might produce a badly conditioned estimated innovations covariance and induce numerical instability. Conditioning improves when you set
ResCond=true
.
Algorithms
In the presence of nonspherical innovations, GLS produces efficient estimates relative to OLS and consistent coefficient covariances, conditional on the innovations covariance. The degree to which
fgls
maintains these properties depends on the accuracy of both the model and estimation of the innovations covariance.Rather than estimate FGLS estimates the usual way,
fgls
uses methods that are faster and more stable, and are applicable to rank-deficient cases.Traditional FGLS methods, such as the Cochrane-Orcutt procedure, use low-order, autoregressive models. These methods, however, estimate parameters in the innovations covariance matrix using OLS, where
fgls
uses maximum likelihood estimation (MLE) [2].
References
[1] Cribari-Neto, F. "Asymptotic Inference Under Heteroskedasticity of Unknown Form." Computational Statistics & Data Analysis. Vol. 45, 2004, pp. 215–233.
[2] Hamilton, James D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.
[3] Judge, G. G., W. E. Griffiths, R. C. Hill, H. Lϋtkepohl, and T. C. Lee. The Theory and Practice of Econometrics. New York, NY: John Wiley & Sons, Inc., 1985.
[4] Kutner, M. H., C. J. Nachtsheim, J. Neter, and W. Li. Applied Linear Statistical Models. 5th ed. New York: McGraw-Hill/Irwin, 2005.
[5] MacKinnon, J. G., and H. White. "Some Heteroskedasticity-Consistent Covariance Matrix Estimators with Improved Finite Sample Properties." Journal of Econometrics. Vol. 29, 1985, pp. 305–325.
[6] White, H. "A Heteroskedasticity-Consistent Covariance Matrix and a Direct Test for Heteroskedasticity." Econometrica. Vol. 48, 1980, pp. 817–838.
Version History
Introduced in R2014bR2022a: fgls
returns estimates in tables when you supply a table of data
If you supply a table of time series data Tbl
,
fgls
returns the following outputs:
In the first position,
fgls
returns the tableCoeffTbl
containing variables for coefficient estimatesCoeff
and standard errorsSE
with rows corresponding to, and labeled as,VarNames
.In the second position,
fgls
returns a table containing the estimated coefficient covariancesCovTbl
with rows and variables corresponding to, and labeled as,VarNames
.
Before R2022a, fgls
returned the numeric outputs in separate
positions of the output when you supplied a table of input data.
Starting in R2022a, if you supply a table of input data, update your code to return all outputs in the first through third output positions.
[CoeffTbl,CovTbl,iterPlots] = fgls(Tbl,Name=Value)
If you request more outputs, fgls
issues an error.
Also, access results by using table indexing. For more details, see Access Data in Tables.
See Also
Functions
Objects
Topics
- Classical Model Misspecification Tests
- Time Series Regression I: Linear Models
- Time Series Regression VI: Residual Diagnostics
- Time Series Regression X: Generalized Least Squares and HAC Estimators
- Autocorrelation and Partial Autocorrelation
- Engle’s ARCH Test
- Nonspherical Models
- Time Series Regression Models
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