infer
Infer residuals of univariate regression model with ARIMA time series errors
Syntax
Description
returns the table or timetable Tbl2
= infer(Mdl
,Tbl1
)Tbl2
containing paths of residuals,
unconditional disturbances, innovation variances inferred from the model
Mdl
and the response data in the input table or timetable
Tbl1
. (since R2023b)
infer
selects the response variable named in
Mdl.SeriesName
or the sole variable in Tbl1
. To
select a different response variable in Tbl1
to infer residuals,
unconditional disturbances, and innovation variances, use the
ResponseVariable
name-value argument.
[___] = infer(___,
specifies options using one or more name-value arguments in
addition to any of the input argument combinations in previous syntaxes.
Name=Value
)infer
returns the output argument combination for the
corresponding input arguments. For example, infer(Mdl,Y,U0=u0,X=Pred)
infers residuals
from the numeric vector of response data Y
with respect to the
regression model with ARIMA errors Mdl
, and specifies the numeric
vector of presample regression model residual data u0
to initialize the
model and the predictor data Pred
for the regression component.
Examples
Infer Vector of Residuals from Regression Model with ARIMA Errors
Infer error model residuals from a simulated path of responses from the following regression model with ARMA(2,1) errors:
where is Gaussian with variance 0.1. Assume the predictors are standard Gaussian random variables. Provide data as numeric arrays.
Create the regression model with ARIMA errors. Simulate responses from the model and two predictor series.
Mdl = regARIMA(Intercept=0,AR={0.5 -0.8},MA=-0.5, ... Beta=[0.1; -0.2],Variance=0.1); rng(1,"twister"); % For reproducibility Pred = randn(100,2); y = simulate(Mdl,100,X=Pred);
Infer and plot the error model residuals. By default, infer
backcasts for the necessary presample unconditional disturbances and sets necessary presample error model residuals to zero.
e = infer(Mdl,y,X=Pred);
figure
plot(e)
title("Inferred Residuals")
e
is a 100-by-1 vector of error model residuals, associated with error model innovations .
Examine Residuals of Estimated Model in Timetable
Since R2023b
Fit a regression model with ARMA(1,1) errors by regressing the US gross domestic product (GDP) growth rate onto consumer price index (CPI) quarterly changes. Examine the error model and regression residuals. Supply a timetable of data and specify the series for the fit.
Load and Transform Data
Load the US macroeconomic data set. Compute the series of GDP quarterly growth rates and CPI quarterly changes.
load Data_USEconModel DTT = price2ret(DataTimeTable,DataVariables="GDP"); DTT.GDPRate = 100*DTT.GDP; DTT.CPIDel = diff(DataTimeTable.CPIAUCSL); T = height(DTT)
T = 248
figure tiledlayout(2,1) nexttile plot(DTT.Time,DTT.GDPRate) title("GDP Rate") ylabel("Percent Growth") nexttile plot(DTT.Time,DTT.CPIDel) title("Index")
The series appear stationary, albeit heteroscedastic.
Prepare Timetable for Estimation
When you plan to supply a timetable, you must ensure it has all the following characteristics:
The selected response variable is numeric and does not contain any missing values.
The timestamps in the
Time
variable are regular, and they are ascending or descending.
Remove all missing values from the timetable.
DTT = rmmissing(DTT); T_DTT = height(DTT)
T_DTT = 248
Because each sample time has an observation for all variables, rmmissing
does not remove any observations.
Determine whether the sampling timestamps have a regular frequency and are sorted.
areTimestampsRegular = isregular(DTT,"quarters")
areTimestampsRegular = logical
0
areTimestampsSorted = issorted(DTT.Time)
areTimestampsSorted = logical
1
areTimestampsRegular = 0
indicates that the timestamps of DTT
are irregular. areTimestampsSorted = 1
indicates that the timestamps are sorted. Macroeconomic series in this example are timestamped at the end of the month. This quality induces an irregularly measured series.
Remedy the time irregularity by shifting all dates to the first day of the quarter.
dt = DTT.Time; dt = dateshift(dt,"start","quarter"); DTT.Time = dt; areTimestampsRegular = isregular(DTT,"quarters")
areTimestampsRegular = logical
1
DTT
is regular.
Create Model Template for Estimation
Suppose that a regression model of CPI quarterly changes onto the GDP rate, with ARMA(1,1) errors, is appropriate.
Create a model template for a regression model with ARMA(1,1) errors template. Specify the response variable name.
Mdl = regARIMA(1,0,1);
Mdl.SeriesName = "GDPRate";
Mdl
is a partially specified regARIMA
object.
Fit Model to Data
Fit a regression model with ARMA(1,1) errors to the data. Specify the entire series GDP rate and CPI quarterly changes series, and specify the predictor variable name.
EstMdl = estimate(Mdl,DTT,PredictorVariables="CPIDel");
Regression with ARMA(1,1) Error Model (Gaussian Distribution): Value StandardError TStatistic PValue ________ _____________ __________ __________ Intercept 0.0162 0.0016077 10.077 6.9994e-24 AR{1} 0.60515 0.089912 6.7305 1.6906e-11 MA{1} -0.16221 0.11051 -1.4678 0.14216 Beta(1) 0.002221 0.00077691 2.8587 0.0042532 Variance 0.000113 7.2753e-06 15.533 2.0838e-54
EstMdl
is a fully specified, estimated regARIMA
object. By default, estimate
backcasts for the required Mdl.P = 1
presample regression model residual and sets the required Mdl.Q = 1
presample error model residual to 0.
Examine Residuals
Infer a timetable of error model and regression residuals for all observations. Specify the predictor variable name.
Tbl2 = infer(EstMdl,DTT,PredictorVariables="CPIDel")
Tbl2=248×6 timetable
Time Interval GDP GDPRate CPIDel GDPRate_ErrorResidual GDPRate_RegressionResidual
_____ ________ ___________ _________ ______ _____________________ __________________________
Q2-47 91 0.00015183 0.015183 0.08 -0.0007572 -0.0011947
Q3-47 92 0.00018374 0.018374 0.76 0.0010863 0.00048617
Q4-47 92 0.000427 0.0427 0.57 0.025116 0.025234
Q1-48 91 0.00025617 0.025617 0.09 -0.0019795 0.0092168
Q2-48 91 0.00028739 0.028739 0.65 0.005197 0.011096
Q3-48 92 0.00026512 0.026512 0.21 0.0039745 0.0098461
Q4-48 92 5.1468e-05 0.0051468 -0.31 -0.015678 -0.010365
Q1-49 90 -0.00021196 -0.021196 -0.14 -0.033356 -0.037085
Q2-49 91 -0.00015576 -0.015576 0.01 -0.014767 -0.031798
Q3-49 92 6.1077e-05 0.0061077 -0.17 0.0071327 -0.0097147
Q4-49 91 -0.00010311 -0.010311 -0.14 -0.019164 -0.0262
Q1-50 91 0.00040675 0.040675 0.03 0.037154 0.024408
Q2-50 91 0.00036908 0.036908 0.24 0.011432 0.020175
Q3-50 91 0.00065211 0.065211 0.46 0.037635 0.04799
Q4-50 91 0.00040718 0.040718 0.64 0.00016008 0.023097
Q1-51 91 0.00053382 0.053382 0.9 0.021232 0.035183
⋮
Tbl2
is a 248-by-6 timetable containing the error model residuals GDPRate_ErrorResidual
, regression residuals GDPRate_RegressionResidual
, and all variables in DTT
.
Separately plot the inferred error model and regression residuals.
Tbl2.GDPRate_Fitted = Tbl2.GDPRate - Tbl2.GDPRate_RegressionResidual; figure h = tiledlayout(2,2); title(h,"Error Model Residuals") nexttile plot(Tbl2.Time,Tbl2.GDPRate_ErrorResidual,'b',Tbl2.Time([1 end]),[0 0],'--r') title("Case Order") nexttile histogram(Tbl2.GDPRate_ErrorResidual) title("Histogram") nexttile plot(Tbl2.GDPRate_ErrorResidual(1:end-1),Tbl2.GDPRate_ErrorResidual(2:end),'o') title("e_{t-1} versus e_t") nexttile plot(Tbl2.GDPRate_Fitted,Tbl2.GDPRate_ErrorResidual,'o') title("Fitted versus e_t")
figure h = tiledlayout(2,2); title(h,"Regression Residuals") nexttile plot(Tbl2.Time,Tbl2.GDPRate_RegressionResidual,'b',Tbl2.Time([1 end]),[0 0],'--r') title("Case Order") nexttile histogram(Tbl2.GDPRate_RegressionResidual) title("Histogram") nexttile plot(Tbl2.GDPRate_RegressionResidual(1:end-1),Tbl2.GDPRate_RegressionResidual(2:end),'o') title("e_{t-1} versus e_t") nexttile plot(Tbl2.GDPRate_Fitted,Tbl2.GDPRate_RegressionResidual,'o') title("Fitted versus e_t")
Compare Model Fits By Using Likelihood Ratio Test
Fit this regression model with ARMA(2,1) errors to simulated data:
where is Gaussian with variance 0.1. Compare the fit to an intercept-only regression model by conducting a likelihood ratio test. Provide response and predictor data in vectors.
Simulate Data
Specify the regression model ARMA(2,1) errors. Simulate responses from the model, and simulate two predictor series from the standard Gaussian distribution.
Mdl0 = regARIMA(Intercept=1,AR={0.5 -0.8},MA=-0.5, ... Beta=[0.1; -0.2],Variance=0.1); rng(1,"twister") % For reproducibility Pred = randn(100,2); y = simulate(Mdl0,100,X=Pred);
y
is a 100-by-1 random response path simulated from Mdl
.
Fit Unrestricted Model
Create an unrestricted model template of a regression model with ARMA(2,1) errors for estimation.
Mdl = regARIMA(2,0,1)
Mdl = regARIMA with properties: Description: "ARMA(2,1) Error Model (Gaussian Distribution)" SeriesName: "Y" Distribution: Name = "Gaussian" Intercept: NaN Beta: [1×0] P: 2 Q: 1 AR: {NaN NaN} at lags [1 2] SAR: {} MA: {NaN} at lag [1] SMA: {} Variance: NaN
The AR coefficients, MA coefficients, and the innovation variance are NaN
values. estimate
estimates those parameters. When Beta
is an empty array, estimate
determines the number of regression coefficients to estimate.
Fit the unrestricted model to the data. Specify the predictor data.
EstMdlUR = estimate(Mdl,y,X=Pred);
Regression with ARMA(2,1) Error Model (Gaussian Distribution): Value StandardError TStatistic PValue ________ _____________ __________ __________ Intercept 1.0167 0.010154 100.13 0 AR{1} 0.64995 0.093794 6.9295 4.2226e-12 AR{2} -0.69174 0.082575 -8.3771 5.4247e-17 MA{1} -0.64508 0.11055 -5.835 5.3796e-09 Beta(1) 0.10866 0.020965 5.183 2.1835e-07 Beta(2) -0.20979 0.022824 -9.1917 3.8679e-20 Variance 0.073117 0.008716 8.3888 4.9121e-17
EstMdlUR
is a fully specified regARIMA
object representing the estimated unrestricted regression model with ARIMA errors.
Fit Restricted Model
The restricted model contains the same error model, but the regression model contains only an intercept. That is, the restricted model imposes two restrictions on the unrestricted model: .
Fit the restricted model to the data.
EstMdlR = estimate(Mdl,y);
ARMA(2,1) Error Model (Gaussian Distribution): Value StandardError TStatistic PValue ________ _____________ __________ __________ Intercept 1.0176 0.024905 40.859 0 AR{1} 0.51541 0.18536 2.7805 0.0054271 AR{2} -0.53359 0.10949 -4.8735 1.0963e-06 MA{1} -0.34923 0.19423 -1.798 0.07218 Variance 0.1445 0.020214 7.1486 8.7671e-13
EstMdlR
is a fully specified regARIMA
object representing the estimated restricted regression model with ARIMA errors.
Compute Residuals and Loglikelihoods
Compute the residual series and loglikelihoods for the estimated models.
[eUR,uUR,~,logLUR] = infer(EstMdlUR,y,X=Pred); [eR,uR,~,logLR] = infer(EstMdlR,y);
eUR
and uUR
are 100-by-1 vectors containing the error model and regression residuals from the unrestricted estimation. loglUR
is the corresponding loglikelihood.
eR
and uR
are 100-by-1 vectors containing the error model and regression residuals from the restricted estimation. loglR
is the corresponding loglikelihood.
Conduct Likelihood Ratio Test
The likelihood ratio test requires the optimized loglikelihoods of the unrestricted and restricted models, and it requires the number of model restrictions (degrees of freedom).
Conduct a likelihood ratio test to determine which model has the better fit to the data.
dof = 2; [h,p] = lratiotest(logLUR,logLR,dof)
h = logical
1
p = 1.6653e-15
The -value is close to zero, which suggests that there is strong evidence to reject the null hypothesis that the data fits the restricted model better than the unrestricted model.
Input Arguments
Y
— Response data yt
numeric column vector | numeric matrix
Response data yt, specified as a
numobs
-by-1 numeric column vector or
numobs
-by-numpaths
numeric matrix.
numObs
is the length of the time series (sample size).
numpaths
is the number of separate, independent paths of response
series.
infer
infers the residuals, unconditional disturbances,
and innovation variances of columns of Y
, which are time series
characterized by Mdl
.
Each row corresponds to a sampling time. The last row contains the latest set of observations.
Each column corresponds to a separate, independent path of response data.
infer
assumes that responses across any row occur
simultaneously.
Data Types: double
Tbl1
— Time series data
table | timetable
Since R2023b
Time series data containing the observed response variable
yt and, optionally, predictor variables
xt for the regression component, specified
as a table or timetable with numvars
variables and
numobs
rows. You can optionally select the response variable or
numpreds
predictor variables by using the
ResponseVariable
or PredictorVariables
name-value arguments, respectively.
Each row is an observation, and measurements in each row occur simultaneously. The
selected response variable is a single path (numobs
-by-1 vector) or
multiple paths (numobs
-by-numpaths
matrix) of
numobs
observations of response data.
Each path (column) of the selected response variable is independent of the other
paths, but path
of all presample and
in-sample variables correspond, for j
=
1,…,j
numpaths
. Each selected predictor variable is a
numobs
-by-1 numeric vector representing one path. The
infer
function includes all predictor variables in the
model when it infers residuals. Variables in Tbl1
represent the
continuation of corresponding variables in Presample
.
If Tbl1
is a timetable, it must represent a sample with a
regular datetime time step (see isregular
), and the datetime vector Tbl1.Time
must be
strictly ascending or descending.
If Tbl1
is a table, the last row contains the latest
observation.
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: infer(Mdl,Y,U0=u0,X=Pred)
infers residuals from the numeric
vector of response data Y
with respect to the regression model with
ARIMA errors Mdl
, and specifies the numeric vector of presample
regression model residual data u0
to initialize the model and the
predictor data Pred
for the regression component.
ResponseVariable
— Response variable yt to select from Tbl1
string scalar | character vector | integer | logical vector
Since R2023b
Response variable yt to select from
Tbl1
containing the response data, specified as one of the
following data types:
String scalar or character vector containing a variable name in
Tbl1.Properties.VariableNames
Variable index (positive integer) to select from
Tbl1.Properties.VariableNames
A logical vector, where
DisturbanceVariable(
selects variablej
) = true
fromj
Tbl1.Properties.VariableNames
The selected variable must be a numeric vector and cannot contain missing values
(NaN
s).
If Tbl1
has one variable, the default specifies that variable.
Otherwise, the default matches the variable to names in
Mdl.SeriesName
.
Example: ResponseVariable="StockRate"
Example: ResponseVariable=[false false true false]
or
ResponseVariable=3
selects the third table variable as the
response variable.
Data Types: double
| logical
| char
| cell
| string
X
— Predictor data
numeric matrix
Predictor data for the model regression component, specified as a numeric matrix
with numpreds
columns. numpreds
is the number of
predictor variables (numel(Mdl.Beta)
). Use X
only when you supply the numeric array of response data Y
.
X
must have at least numobs
rows. If the
number of rows of X
exceeds numobs
,
infer
uses only the latest observations.
infer
does not use the regression component in the
presample period.
Columns of X
are separate predictor variables.
infer
applies X
to each path; that
is, X
represents one path of observed predictors.
By default, infer
excludes the regression component,
regardless of its presence in Mdl
.
Data Types: double
PredictorVariables
— Predictor variables xt to select from Tbl1
string vector | cell vector of character vectors | vector of integers | logical vector
Predictor variables xt to select from
Tbl1
containing the predictor data for the model regression
component, specified as one of the following data types:
String vector or cell vector of character vectors containing
numpreds
variable names inTbl1.Properties.VariableNames
A vector of unique indices (positive integers) of variables to select from
Tbl1.Properties.VariableNames
A logical vector, where
PredictorVariables(
selects variablej
) = true
fromj
Tbl1.Properties.VariableNames
The selected variables must be numeric vectors and cannot contain missing values (NaN
s).
By default, infer
excludes the regression component, regardless of its presence in Mdl
.
Example: PredictorVariables=["M1SL" "TB3MS" "UNRATE"]
Example: PredictorVariables=[true false true false]
or PredictorVariable=[1 3]
selects the first and third table variables to supply the predictor data.
Data Types: double
| logical
| char
| cell
| string
E0
— Presample error model residual data et
numeric column vector | numeric matrix
Presample error model residual data et
to initialize the error model, specified as a numpreobs
-by-1
numeric column vector or a
numpreobs
-by-numprepaths
numeric matrix. Use
E0
only when you supply the numeric array of response data
Y
.
Each row is a presample observation (sampling time), and measurements in each row
occur simultaneously. The last row contains the latest presample observation.
numpreobs
must be at least Mdl.Q
to initialize
the moving average (MA) component of the error model. If numpreobs
is larger than required, infer
uses the latest required
number of observations only.
Columns of E0
are separate, independent presample paths. The
following conditions apply:
If
E0
is a column vector, it represents a single residual path.infer
applies it to each output path.If
E0
is a matrix, each column represents a presample residual path.infer
appliesE0(:,
to initialize pathj
)j
.numprepaths
must be at leastnumpaths
. Ifnumprepaths
>numpaths
,infer
uses the firstsize(Y,2)
columns only.infer
assumes each column ofE0
has a mean of zero.
By default, infer
sets the necessary presample
disturbances to zero.
Data Types: double
U0
— Presample regression residual data
numeric column vector | numeric matrix
Presample regression residual data, associated with the unconditional disturbances
ut, to initialize the error model,
specified as a numpreobs
-by-1 numeric column vector or a
numpreobs
-by-numprepaths
numeric matrix. Use
U0
only when you supply the numeric array of response data
Y
.
Each row is a presample observation (sampling time), and measurements in each row
occur simultaneously. The last row contains the latest presample observation.
numpreobs
must be at least Mdl.P
to initialize
the error model autoregressive (AR) component. If numpreobs
is
larger than required, infer
uses the latest required
observations only.
Columns of U0
are separate, independent presample paths. The
following conditions apply:
If
U0
is a column vector, it represents a single path.infer
applies it to each path.If
U0
is a matrix, each column represents a presample path.infer
appliesU0(:,
to initialize pathj
)j
.numprepaths
must be at leastnumpaths
. Ifnumprepaths
>numpaths
,infer
uses the firstsize(Z,2)
columns only.
By default, infer
backcasts for necessary presample
unconditional disturbances.
Data Types: double
Presample
— Presample data
table | timetable
Since R2023b
Presample data containing paths of error model residual
et or regression residual series to
initialize the model, specified as a table or timetable, the same type as
Tbl1
, with numprevars
variables and
numpreobs
rows. Regression residuals are associated with the
unconditional disturbances ut. Use
Presample
only when you supply a table or timetable of data
Tbl1
.
Each selected variable is a single path (numpreobs
-by-1 vector)
or multiple paths (numpreobs
-by-numprepaths
matrix) of numpreobs
observations representing the presample of the
error model or regression residual series for ResponseVariable
,
the selected response variable in Tbl1
.
Each row is a presample observation, and measurements in each row occur
simultaneously. numpreobs
must be one of the following values:
At least
Mdl.P
whenPresample
provides only presample regression residualsAt least
Mdl.Q
whenPresample
provides only presample error model residualsAt least
max([Mdl.P Mdl.Q])
otherwise
If you supply more rows than necessary, infer
uses the
latest required number of observations only.
When Presample
provides presample residuals,
infer
assumes each presample error model residual path
has a mean of zero.
If Presample
is a timetable, all the following conditions
must be true:
Presample
must represent a sample with a regular datetime time step (seeisregular
).The inputs
Tbl1
andPresample
must be consistent in time such thatPresample
immediately precedesTbl1
with respect to the sampling frequency and order.The datetime vector of sample timestamps
Presample.Time
must be ascending or descending.
If Presample
is a table, the last row contains the latest
presample observation.
By default, infer
backcasts for necessary presample
regression residuals and sets necessary presample error model residuals to
zero.
If you specify the Presample
, you must specify the presample
error model or regression residual name by using the
PresampleInnovationVariable
or
PresampleRegressionDisturbanceVariable
name-value
argument.
PresampleInnovationVariable
— Error model residual et to select from Presample
string scalar | character vector | integer | logical vector
Since R2023b
Error model residual variable et to
select from Presample
containing the presample error model residual
data, specified as one of the following data types:
String scalar or character vector containing the variable name to select from
Presample.Properties.VariableNames
Variable index (positive integer) to select from
Presample.Properties.VariableNames
A logical vector, where
PresampleInnovationVariable(
selects variablej
) = true
fromj
Presample.Properties.VariableNames
The selected variable must be a numeric vector and cannot contain missing values
(NaN
s).
If you specify presample error model residual data by using the
Presample
name-value argument, you must specify
PresampleInnovationVariable
.
Example: PresampleInnovationVariable="GDP_Z"
Example: PresampleInnovationVariable=[false false true false]
or
PresampleInnovationVariable=3
selects the third table variable
for presample error model residual data.
Data Types: double
| logical
| char
| cell
| string
PresampleRegressionDistrubanceVariable
— Regression model residual variable to select from Presample
string scalar | character vector | integer | logical vector
Since R2023b
Regression model residual variable, associated with unconditional disturbances
ut, to select from
Presample
containing data for the presample regression model
residuals, specified as one of the following data types:
String scalar or character vector containing a variable name in
Presample.Properties.VariableNames
Variable index (positive integer) to select from
Presample.Properties.VariableNames
A logical vector, where
PresampleRegressionDistrubanceVariable(
selects variablej
) = true
fromj
Presample.Properties.VariableNames
The selected variable must be a numeric vector and cannot contain missing values
(NaN
s).
If you specify presample regression model residual data by using the
Presample
name-value argument, you must specify
PresampleRegressionDistrubanceVariable
.
Example: PresampleRegressionDistrubanceVariable="StockRateU"
Example: PresampleRegressionDistrubanceVariable=[false false true
false]
or PresampleRegressionDistrubanceVariable=3
selects the third table variable as the presample regression model residual
data.
Data Types: double
| logical
| char
| cell
| string
Note
NaN
values inY
,X
,E0
andU0
indicate missing values.infer
removes missing values from specified data by listwise deletion.For the presample,
infer
horizontally concatenates the possibly jagged arraysE0
andU0
with respect to the last rows, and then it removes any row of the concatenated matrix containing at least oneNaN
.For in-sample data,
infer
horizontally concatenates the possibly jagged arraysY
andX
, and then it removes any row of the concatenated matrix containing at least oneNaN
.
This type of data reduction reduces the effective sample size and can create an irregular time series.
For numeric data inputs,
infer
assumes that you synchronize the presample data such that the latest observations occur simultaneously.infer
issues an error when any table or timetable input contains missing values.All predictor variables (columns) in
X
are associated with each input response series to producenumpaths
output series.
Output Arguments
E
— Inferred error model residuals et
numeric matrix
Inferred error model residuals et,
returned as a numobs
-by-numpaths
numeric matrix.
infer
returns E
only when you supply
the input Y
.
E(
is
the path j
,k
)
error model residual of time
k
j
; it is the error model residual associated with response
Y(
.j
,k
)
Inferred residuals are
is row t of the inferred unconditional disturbances
U
,
ϕj is composite
autoregressive coefficient j, and
θk is composite moving
average coefficient k.
U
— Inferred regression residuals
numeric matrix
Inferred regression residuals associated with the unconditional disturbances
ut, returned as a
numobs
-by-numpaths
numeric matrix.
infer
returns V
only when you supply
the input Y
.
U(
is
the path j
,k
)
regression model residual of
time k
j
; it is the regression model residual associated with
response
Y(
.j
,k
)
Inferred unconditional disturbances are
yt is row
t of the response data Y
,
xt is row
t of the predictor data X
,
c is the model intercept Mdl.Intercept
, and
β is the vector of regression coefficients
Mdl.Beta
.
V
— Inferred innovation variances
numeric matrix
Inferred innovation variances, returned as a
numobs
-by-numpaths
numeric matrix.
infer
returns V
only when you supply
the input Y
. All elements in V
are equal to
Mdl.Variance
.
Tbl2
— Inferred error model residual et and regression residual paths
table | timetable
Since R2023b
Inferred error model residual et and
regression residual paths, returned as a table or timetable, the same data type as
Tbl1
. infer
returns
Tbl2
only when you supply the input Tbl1
.
Regression residuals are associated with the unconditional disturbances
ut.
Tbl2
contains the following variables:
The inferred error model residual paths, which are in a
numobs
-by-numpaths
numeric matrix, with rows representing observations and columns representing independent paths. Each path corresponds to the input response path inTbl1
and represents the continuation of the corresponding presample error model residual path inPresample
.infer
names the inferred residual variable inTbl2
, whereresponseName
_ErrorResidual
isresponseName
Mdl.SeriesName
. For example, ifMdl.SeriesName
isStockReturns
,Tbl2
contains a variable for the corresponding inferred error model residual paths with the nameStockReturns_ErrorResidual
.The inferred regression residual paths, which are in a
numobs
-by-numpaths
numeric matrix, with rows representing observations and columns representing independent paths. Each path represents the continuation of the corresponding path of presample regression residuals inPresample
.infer
names the inferred regression residual variable inTbl2
, whereresponseName
_RegressionResidual
isresponseName
Mdl.SeriesName
. For example, ifMdl.SeriesName
isStockReturns
,Tbl2
contains a variable for the corresponding inferred regression residual paths with the nameStockReturns_RegressionResidual
.All variables
Tbl1
.
If Tbl1
is a timetable, row times of Tbl1
and Tbl2
are equal.
Tbl2
does not include a variable containing inferred paths of
innovation variances. To create such a variable, enter
Tbl2.
.responseName
_Variance =
Mdl.Variance*ones(size(Tbl2));
logL
— Loglikelihood objective function values
numeric scalar | numeric vector
References
[1] Box, George E. P., Gwilym M. Jenkins, and Gregory C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.
[2] Davidson, R., and J. G. MacKinnon. Econometric Theory and Methods. Oxford, UK: Oxford University Press, 2004.
[3] Enders, Walter. Applied Econometric Time Series. Hoboken, NJ: John Wiley & Sons, Inc., 1995.
[4] Hamilton, James D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.
[5] Pankratz, A. Forecasting with Dynamic Regression Models. John Wiley & Sons, Inc., 1991.
[6] Tsay, R. S. Analysis of Financial Time Series. 2nd ed. Hoboken, NJ: John Wiley & Sons, Inc., 2005.
Version History
Introduced in R2013bR2023b: infer
accepts input data in tables and timetables
In addition to accepting input data (in-sample and presample data) in numeric arrays,
infer
accepts input data in tables or regular timetables. When
you supply data in a table or timetable, the following conditions apply:
infer
chooses the default in-sample response series on which to operate, but you can use the specified optional name-value argument to select a different series.If you specify optional presample error model residual or regression model residual data to initialize the model, you must also specify the appropriate presample variable names.
infer
returns results in a table or timetable.
Name-value arguments to support tabular workflows include:
ResponseVariable
specifies the name of the response series to select from the input data, from which residuals are inferred.PredictorVariables
specifies the names of the predictor series to select from the input data for a model regression component.Presample
specifies the input table or timetable of presample regression residual or error model residual data.PresampleInnovationVariable
specifies the name of the error model residual series to select fromPresample
.PresampleRegressionDisturbanceVariable
specifies the name of the regression residual series to select fromPresample
.
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