infer
Infer vector autoregression model (VAR) innovations
Description
returns the table or timetable Tbl2
= infer(Mdl
,Tbl1
)Tbl2
containing the multivariate
residuals from evaluating the fully specified VAR(p) model
Mdl
at the response variables in the table or timetable of
data Tbl1
. (since R2022b)
infer
selects the variables in Mdl.SeriesNames
or all variables in Tbl1
. To select different response variables in Tbl1
at which to evaluate the model, use the ResponseVariables
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,Y0=PS,X=Exo)
computes the
residuals of the VAR(p) model Mdl
at the
matrix of response data Y
, and specifies the matrix of presample
response data PS
and the matrix of exogenous predictor data
Exo
.
Supply all input data using the same data type. Specifically:
If you specify the numeric matrix
Y
, optional data sets must be numeric arrays and you must use the appropriate name-value argument. For example, to specify a presample, set theY0
name-value argument to a numeric matrix of presample data.If you specify the table or timetable
Tbl1
, optional data sets must be tables or timetables, respectively, and you must use the appropriate name-value argument. For example, to specify a presample, set thePresample
name-value argument to a table or timetable of presample data.
Examples
Infer VAR(4) Model Innovations From Matrix of Response Data
Fit a VAR(4) model to the consumer price index (CPI) and unemployment rate data in a matrix. Then, infer the model innovations (residuals) from the estimated model.
Load the Data_USEconModel
data set.
load Data_USEconModel
Plot the two series on separate plots.
figure plot(DataTimeTable.Time,DataTimeTable.CPIAUCSL) title("Consumer Price Index") ylabel("Index") xlabel("Date")
figure plot(DataTimeTable.Time,DataTimeTable.UNRATE) title("Unemployment Rate") ylabel("Percent") xlabel("Date")
Stabilize the CPI by converting it to a series of growth rates. Synchronize the two series by removing the first observation from the unemployment rate series.
rcpi = price2ret(DataTimeTable.CPIAUCSL); unrate = DataTimeTable.UNRATE(2:end);
Create a default VAR(4) model by using the shorthand syntax.
Mdl = varm(2,4);
Estimate the model using the entire data set.
EstMdl = estimate(Mdl,[rcpi unrate]);
EstMdl
is a fully specified, estimated varm
model object.
Infer innovations from the estimated model. Supply the same response data that the model was fit to as a numeric matrix.
E = infer(EstMdl,[rcpi unrate]);
E
is a 241-by-2 matrix of inferred innovations. The first and second columns contain the residuals corresponding to the CPI growth rate and unemployment rate, respectively.
Alternatively, you can return residuals when you call estimate
by supplying an output variable in the fourth position.
Plot the residuals on separate plots. Synchronize the residuals with the dates by removing any missing observations from the data and removing the first Mdl.P
dates.
idx = all(~isnan([rcpi unrate]),2); datesr = DataTimeTable.Time(idx); figure plot(datesr((Mdl.P + 1):end),E(:,1)); ylabel("Consumer Price Index") xlabel("Date") title("Residual Plot") hold on yline(0,"r--"); hold off
figure plot(datesr((Mdl.P + 1):end),E(:,2)) ylabel("Unemployment Rate") xlabel("Date") title("Residual Plot") hold on yline(0,"r--"); hold off
The residuals corresponding to the CPI growth rate exhibit heteroscedasticity because the series appears to cycle through periods of higher and lower variance.
Infer VAR(4) Model Innovations from Timetable of Response Data
Since R2022b
Fit a VAR(4) model to the consumer price index (CPI) and unemployment rate data in a timetable. Then, infer the model innovations (residuals) from the estimated model.
Load and Preprocess Data
Load the Data_USEconModel
data set. Compute the CPI growth rate. Because the growth rate calculation consumes the earliest observation, include the rate variable in the timetable by prepending the series with NaN
.
load Data_USEconModel
DataTimeTable.RCPI = [NaN; price2ret(DataTimeTable.CPIAUCSL)];
numobs = height(DataTimeTable)
numobs = 249
Prepare Timetable for Estimation
When you plan to supply a timetable directly to estimate, you must ensure it has all the following characteristics:
All selected response variables are numeric and do 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 table, relative to the CPI rate (RCPI
) and unemployment rate (UNRATE
) series.
varnames = ["RCPI" "UNRATE"]; DTT = rmmissing(DataTimeTable,DataVariables=varnames); numobs = height(DTT)
numobs = 245
rmmissing
removes the four initial missing observations from the DataTimeTable
to create a sub-table DTT
. The variables RCPI
and UNRATE
of DTT
do not have any missing 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 with respect to time.
Create Model Template for Estimation
Create a default VAR(4) model by using the shorthand syntax. Specify the response variable names.
Mdl = varm(2,4); Mdl.SeriesNames = varnames;
Fit Model to Data
Estimate the model. Pass the entire timetable DTT
. By default, estimate
selects the response variables in Mdl.SeriesNames
to fit to the model. Alternatively, you can use the ResponseVariables
name-value argument.
EstMdl = estimate(Mdl,DTT);
Compute Residuals
Infer innovations from the estimated model. Supply the same response data that the model was fit to as a timetable. By default, infer
selects the variables to use from EstMdl.SeriesNames
.
Tbl = infer(EstMdl,DTT); head(Tbl)
Time COE CPIAUCSL FEDFUNDS GCE GDP GDPDEF GPDI GS10 HOANBS M1SL M2SL PCEC TB3MS UNRATE RCPI RCPI_Residuals UNRATE_Residuals _____ _____ ________ ________ ____ _____ ______ ____ ____ ______ ____ ____ _____ _____ ______ __________ ______________ ________________ Q1-49 144.1 23.91 NaN 45.6 270 16.531 40.9 NaN 53.961 NaN NaN 177 1.17 5 -0.0058382 -0.013422 0.64674 Q2-49 141.9 23.92 NaN 47.3 266.2 16.35 34 NaN 53.058 NaN NaN 178.6 1.17 6.2 0.00041815 0.0051673 0.6439 Q3-49 141 23.75 NaN 47.2 267.7 16.256 37.3 NaN 52.501 NaN NaN 178 1.07 6.6 -0.0071324 0.0030175 -0.099092 Q4-49 140.5 23.61 NaN 46.6 265.2 16.272 35.2 NaN 52.291 NaN NaN 180.4 1.1 6.6 -0.0059122 -0.001196 -0.0066535 Q1-50 144.6 23.64 NaN 45.6 275.2 16.222 44.4 NaN 52.696 NaN NaN 183.1 1.12 6.3 0.0012698 0.0024607 -0.013354 Q2-50 150.6 23.88 NaN 46.1 284.6 16.286 49.9 NaN 53.997 NaN NaN 187 1.15 5.4 0.010101 0.010823 -0.53098 Q3-50 159 24.34 NaN 45.9 302 16.63 56.1 NaN 55.7 NaN NaN 200.7 1.3 4.4 0.01908 0.012566 -0.38177 Q4-50 166.9 24.98 NaN 49.5 313.4 16.95 65.9 NaN 56.213 NaN NaN 198.1 1.34 4.3 0.025954 0.010998 0.50761
size(Tbl)
ans = 1×2
241 17
Tbl
is a 241-by-17 timetable of variables in DTT
and estimated model residuals, RCPI_Residuals
and UNRATE_Residuals
.
Alternatively, you can return residuals when you call estimate
by supplying an output variable in the fourth position.
Infer Innovations from Model Containing Regression Component
Since R2022b
Estimate a VAR(4) model of the consumer price index (CPI), the unemployment rate, and the gross domestic product (GDP). Include a linear regression component containing the current quarter and the last four quarters of government consumption expenditures and investment (GCE). Infer model innovations.
Load the Data_USEconModel
data set. Compute the real GDP.
load Data_USEconModel
DataTimeTable.RGDP = DataTimeTable.GDP./DataTimeTable.GDPDEF*100;
Plot all variables on separate plots.
figure tiledlayout(2,2) nexttile plot(DataTimeTable.Time,DataTimeTable.CPIAUCSL); ylabel("Index") title("Consumer Price Index") nexttile plot(DataTimeTable.Time,DataTimeTable.UNRATE); ylabel("Percent") title("Unemployment Rate") nexttile plot(DataTimeTable.Time,DataTimeTable.RGDP); ylabel("Output") title("Real Gross Domestic Product") nexttile plot(DataTimeTable.Time,DataTimeTable.GCE); ylabel("Billions of $") title("Government Expenditures")
Stabilize the CPI, GDP, and GCE by converting each to a series of growth rates. Synchronize the unemployment rate series with the others by removing its first observation.
varnames = ["CPIAUCSL" "RGDP" "GCE"]; DTT = varfun(@price2ret,DataTimeTable,InputVariables=varnames); DTT.Properties.VariableNames = varnames; DTT.UNRATE = DataTimeTable.UNRATE(2:end);
Make the time base regular.
dt = DTT.Time; dt = dateshift(dt,"start","quarter"); DTT.Time = dt;
Expand the GCE rate series to a matrix that includes the first lagged series through the fourth lag series.
RGCELags = lagmatrix(DTT,1:4,DataVariables="GCE");
DTT = [DTT RGCELags];
DTT = rmmissing(DTT);
Create a default VAR(4) model by using the shorthand syntax. Specify the response variable names.
Mdl = varm(3,4); Mdl.SeriesNames = ["CPIAUCSL" "UNRATE" "RGDP"];
Estimate the model using the entire sample. Specify the GCE and its lags as exogenous predictor data for the regression component.
prednames = contains(DTT.Properties.VariableNames,"GCE");
EstMdl = estimate(Mdl,DTT,PredictorVariables=prednames);
Infer innovations from the estimated model. Supply the predictor data. Return the loglikelihood objective function value.
[Tbl,logL] = infer(EstMdl,DTT,PredictorVariables=prednames); size(Tbl)
ans = 1×2
240 11
head(Tbl)
Time CPIAUCSL RGDP GCE UNRATE Lag1GCE Lag2GCE Lag3GCE Lag4GCE CPIAUCSL_Residuals UNRATE_Residuals RGDP_Residuals _____ __________ __________ __________ ______ __________ __________ __________ __________ __________________ ________________ ______________ Q1-49 0.00041815 -0.0031645 0.036603 6.2 0.047147 0.04948 0.04193 0.054347 0.0053457 0.6564 -0.0053201 Q2-49 -0.0071324 0.011385 -0.0021164 6.6 0.036603 0.047147 0.04948 0.04193 0.0088626 -0.034796 0.010153 Q3-49 -0.0059122 -0.010366 -0.012793 6.6 -0.0021164 0.036603 0.047147 0.04948 0.0029402 0.11695 -0.02318 Q4-49 0.0012698 0.040091 -0.021693 6.3 -0.012793 -0.0021164 0.036603 0.047147 0.0040774 -0.2343 0.026583 Q1-50 0.010101 0.029649 0.010905 5.4 -0.021693 -0.012793 -0.0021164 0.036603 0.0046233 -0.18043 0.0091538 Q2-50 0.01908 0.03844 -0.0043478 4.4 0.010905 -0.021693 -0.012793 -0.0021164 0.015141 -0.34049 0.019797 Q3-50 0.025954 0.017994 0.075508 4.3 -0.0043478 0.010905 -0.021693 -0.012793 0.0041785 0.87368 -0.011263 Q4-50 0.035395 0.01197 0.14807 3.4 0.075508 -0.0043478 0.010905 -0.021693 0.011772 -0.49694 -0.0044563
logL
logL = 1.7056e+03
Tbl
is a 240-by-11 timetable of data and inferred innovations from the estimated model (residuals).
Plot the residuals on separate plots.
idx = endsWith(Tbl.Properties.VariableNames,"_Residuals"); resvars = Tbl.Properties.VariableNames(idx); titles = "Residuals: " + EstMdl.SeriesNames; figure tiledlayout(2,2) for j = 1:Mdl.NumSeries nexttile plot(Tbl.Time,Tbl{:,resvars(j)}); xlabel("Date"); title(titles(j)); hold on yline(0,"r--"); hold off end
The residuals corresponding to the CPI and GDP growth rates exhibit heteroscedasticity because the CPI series appears to cycle through periods of higher and lower variance. Also, the first half of the GDP series seems to have higher variance than the latter half.
Input Arguments
Y
— Response data
numeric matrix | numeric array
Response data, specified as a
numobs
-by-numseries
numeric matrix or a
numobs
-by-numseries
-by-numpaths
numeric array.
numobs
is the sample size. numseries
is the
number of response series (Mdl.NumSeries
).
numpaths
is the number of response paths.
Rows correspond to observations, and the last row contains the latest observation.
Y
represents the continuation of the presample response series in
Y0
.
Columns must correspond to the response variable names in
Mdl.SeriesNames
.
Pages correspond to separate, independent numseries
-dimensional
paths. Among all pages, responses in a particular row occur at the same time.
Data Types: double
Tbl1
— Time series data
table | timetable
Since R2022b
Time series data containing observed response variables
yt and, optionally, predictor
variables xt for a model with a regression
component, specified as a table or timetable with numvars
variables
and numobs
rows.
Each selected response variable is a
numobs
-by-numpaths
numeric matrix, and each
selected predictor variable is a numeric vector. Each row is an observation, and
measurements in each row occur simultaneously. You can optionally specify
numseries
response variables by using the
ResponseVariables
name-value argument, and you can specify
numpreds
predictor variables by using the
PredictorVariables
name-value argument.
Paths (columns) within a particular response variable are independent, but path
of all variables correspond, for
j
=
1,…,j
numpaths
.
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 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,Y0=PS,X=Exo)
computes the residuals of the
VAR(p) model Mdl
at the matrix of response
data Y
, and specifies the matrix of presample response data
PS
and the matrix of exogenous predictor data
Exo
.
ResponseVariables
— Variables to select from Tbl1
to treat as response variables yt
string vector | cell vector of character vectors | vector of integers | logical vector
Since R2022b
Variables to select from Tbl1
to treat as response variables
yt, specified as one of the following
data types:
String vector or cell vector of character vectors containing
numseries
variable names inTbl1.Properties.VariableNames
A length
numseries
vector of unique indices (integers) of variables to select fromTbl1.Properties.VariableNames
A length
numvars
logical vector, whereResponseVariables(
selects variablej
) = true
fromj
Tbl1.Properties.VariableNames
, andsum(ResponseVariables)
isnumseries
The selected variables must be numeric vectors (single path) or matrices (columns
represent multiple independent paths) of the same width, and cannot contain missing
values (NaN
).
If the number of variables in Tbl1
matches
Mdl.NumSeries
, the default specifies all variables in
Tbl1
. If the number of variables in Tbl1
exceeds Mdl.NumSeries
, the default matches variables in
Tbl1
to names in Mdl.SeriesNames
.
Example: ResponseVariables=["GDP" "CPI"]
Example: ResponseVariables=[true false true false]
or
ResponseVariable=[1 3]
selects the first and third table
variables as the response variables.
Data Types: double
| logical
| char
| cell
| string
Y0
— Presample responses
numeric matrix | numeric array
Presample responses that provide initial values for the model
Mdl
, specified as a
numpreobs
-by-numseries
numeric matrix or a
numpreobs
-by-numseries
-by-numprepaths
numeric array. Use Y0
only when you supply a numeric array of
response data Y
.
numpreobs
is the number of presample observations.
numprepaths
is the number of presample response paths.
Each row is a presample observation, and measurements in each row, among all pages,
occur simultaneously. The last row contains the latest presample observation.
Y0
must have at least Mdl.P
rows. If you
supply more rows than necessary, infer
uses the latest
Mdl.P
observations only.
Each column corresponds to the response series associated with the respective response
series in Y
.
Pages correspond to separate, independent paths.
If
Y0
is a matrix,infer
applies it to each path (page) inY
. Therefore, all paths inY
derive from common initial conditions.Otherwise,
infer
appliesY0(:,:,
toj
)Y(:,:,
.j
)Y0
must have at leastnumpaths
pages, andinfer
uses only the firstnumpaths
pages.
By default, infer
uses the first Mdl.P
observations, for example, Y(1:Mdl.P,:)
, as a presample. This action
reduces the effective sample size.
Data Types: double
Presample
— Presample data
table | timetable
Since R2022b
Presample data that provides initial values for the model Mdl
,
specified as a table or timetable, the same type as Tbl1
, with
numprevars
variables and numpreobs
rows.
Each row is a presample observation, and measurements in each row, among all paths,
occur simultaneously. numpreobs
must be at least
Mdl.P
. If you supply more rows than necessary,
infer
uses the latest Mdl.P
observations only.
Each variable is a numpreobs
-by-numprepaths
numeric matrix. Variables correspond to the response series associated with the
respective response variable in Tbl1
. To control presample variable
selection, see the optional PresampleResponseVariables
name-value
argument.
For each variable, columns are separate, independent paths.
If variables are vectors,
infer
applies them to each path inTbl1
to produce the corresponding residuals inTbl2
. Therefore, all response paths derive from common initial conditions.Otherwise, for each variable
and each pathResponseK
,j
infer
appliesPresample.
to produceResponseK
(:,j
)Tbl2.
. Variables must have at leastResponseK
(:,j
)numpaths
columns, andinfer
uses only the firstnumpaths
columns.
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
uses the first or earliest
Mdl.P
observations in Tbl1
as a presample,
and then it fits the model to the remaining numobs – Mdl.P
observations. This action reduces the effective sample size.
PresampleResponseVariables
— Variables to select from Presample
to use for presample response data
string vector | cell vector of character vectors | vector of integers | logical vector
Since R2022b
Variables to select from Presample
to use
for presample data, specified as one of the
following data types:
String vector or cell vector of character vectors containing
numseries
variable names inPresample.Properties.VariableNames
A length
numseries
vector of unique indices (integers) of variables to select fromPresample.Properties.VariableNames
A length
numvars
logical vector, wherePresampleResponseVariables(
selects variablej
) = true
fromj
Presample.Properties.VariableNames
, andsum(PresampleResponseVariables)
isnumseries
The selected variables must be numeric vectors (single path)
or matrices (columns represent multiple independent
paths) of the same width, and cannot contain missing
values (NaN
).
PresampleResponseNames
does not need to
contain the same names as in
Tbl1
;
infer
uses the data in
selected variable
PresampleResponseVariables(
as a presample for the response variable
corresponding to
j
)ResponseVariables(
.j
)
The default specifies the same response variables as those
selected from Tbl1
(see
ResponseVariables
).
Example: PresampleResponseVariables=["GDP"
"CPI"]
Example: PresampleResponseVariables=[true false true
false]
or
PresampleResponseVariable=[1 3]
selects the first and third table variables for
presample data.
Data Types: double
| logical
| char
| cell
| string
X
— Predictor data xt
numeric matrix
Predictor data xt for the regression
component in the model, specified as a numeric matrix containing
numpreds
columns. Use X
only when you supply a
numeric array of response data Y
.
numpreds
is the number of predictor variables
(size(Mdl.Beta,2)
).
Each row corresponds to an observation, and measurements in each row occur
simultaneously. The last row contains the latest observation. X
must
have at least as many observations as Y
. If you supply more rows
than necessary, infer
uses only the latest observations.
infer
does not use the regression component in the
presample period.
If you specify a numeric array for a presample by using
Y0
,X
must have at leastnumobs
rows (seeY
).Otherwise,
X
must have at leastnumobs
–Mdl.P
observations to account for the default presample removal fromY
.
Each column is an individual predictor variable. All predictor variables are present in the regression component of each response equation.
infer
applies X
to each path (page) in
Y
; 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
— Variables to select from Tbl1
to treat as exogenous predictor variables xt
string vector | cell vector of character vectors | vector of integers | logical vector
Since R2022b
Variables to select from Tbl1
to treat as exogenous predictor variables
xt, specified as one of the following data types:
String vector or cell vector of character vectors containing
numpreds
variable names inTbl1.Properties.VariableNames
A length
numpreds
vector of unique indices (integers) of variables to select fromTbl1.Properties.VariableNames
A length
numvars
logical vector, wherePredictorVariables(
selects variablej
) = true
fromj
Tbl1.Properties.VariableNames
, andsum(PredictorVariables)
isnumpreds
The selected variables must be numeric vectors and cannot contain missing values
(NaN
).
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
Note
NaN
values inY
,Y0
, andX
indicate missing values.infer
removes missing values from the data by list-wise deletion.If
Y
is a 3-D array, theninfer
horizontally concatenates the pages ofY
to form anumobs
-by-(numpaths*numseries + numpreds)
matrix.If a regression component is present, then
infer
horizontally concatenatesX
toY
to form anumobs
-by-numpaths*numseries + 1
matrix.infer
assumes that the last rows of each series occur at the same time.infer
removes any row that contains at least oneNaN
from the concatenated data.infer
applies steps 1 and 3 to the presample paths inY0
.
This process ensures that the inferred output innovations of each path are the same size and are based on the same observation times. In the case of missing observations, the results obtained from multiple paths of
Y
can differ from the results obtained from each path individually.This type of data reduction reduces the effective sample size.
infer
issues an error when any table or timetable input contains missing values.
Output Arguments
E
— Inferred multivariate innovations series
numeric matrix | numeric array
Inferred multivariate innovations series, returned as either a numeric matrix, or as a
numeric array that contains columns and pages corresponding to Y
.
infer
returns E
only when you supply a
matrix of response data Y
.
If you specify
Y0
, thenE
hasnumobs
rows (seeY
).Otherwise,
E
hasnumobs
–Mdl.P
rows to account for the presample removal.
Tbl2
— Inferred multivariate innovations series
table | timetable
Since R2022b
Inferred multivariate innovations series and other variables, returned as a table or
timetable, the same data type as Tbl1
.
infer
returns Tbl2
only when you
supply the input Tbl1
.
Tbl2
contains the inferred innovation paths E
from evaluating the model Mdl
at the paths of selected response
variables Y
, and it contains all variables in
Tbl1
. infer
names the innovation
variable corresponding to variable
in ResponseJ
Tbl1
. For example, if one
of the selected response variables for estimation in ResponseJ
_ResidualsTbl1
is
GDP
, Tbl2
contains a variable for the
residuals in the response equation of GDP
with the name
GDP_Residuals
.
If you specify presample response data, Tbl2
and
Tbl1
have the same number of rows, and their rows correspond.
Otherwise, because infer
removes initial observations from
Tbl1
for the required presample by default,
Tbl2
has numobs – Mdl.P
rows to account for
that removal.
If Tbl1
is a timetable, Tbl1
and
Tbl2
have the same row order, either ascending or
descending.
logL
— Loglikelihood objective function value
numeric scalar | numeric vector
Loglikelihood objective function value, returned as a numeric scalar or a
numpaths
-element numeric vector.
logL(
corresponds to the
response path in j
)Y(:,:,
or the path
(column) j
)
of the selected response
variables of j
Tbl1
.
Algorithms
Suppose Y
, Y0
, and X
are the
response, presample response, and predictor data specified by the numeric data inputs in
Y
, Y0
, and X
, or the
selected variables from the input tables or timetables Tbl1
and
Presample
.
infer
infers innovations by evaluating the VAR modelMdl
, specifically,infer
uses this process to determine the time origin t0 of models that include linear time trends.If you do not specify
Y0
, then t0 = 0.Otherwise,
infer
sets t0 tosize(Y0,1)
–Mdl.P
. Therefore, the times in the trend component are t = t0 + 1, t0 + 2,..., t0 +numobs
, wherenumobs
is the effective sample size (size(Y,1)
afterinfer
removes missing values). This convention is consistent with the default behavior of model estimation in whichestimate
removes the firstMdl.P
responses, reducing the effective sample size. Althoughinfer
explicitly uses the firstMdl.P
presample responses inY0
to initialize the model, the total number of observations inY0
andY
(excluding missing values) determines t0.
References
[1] Hamilton, James D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.
[2] Johansen, S. Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. Oxford: Oxford University Press, 1995.
[3] Juselius, K. The Cointegrated VAR Model. Oxford: Oxford University Press, 2006.
[4] Lütkepohl, H. New Introduction to Multiple Time Series Analysis. Berlin: Springer, 2005.
Version History
Introduced in R2017aR2022b: infer
accepts input data in tables and timetables, and return results in tables and timetables
In addition to accepting input data in numeric arrays,
infer
accepts input data in tables and timetables. infer
chooses default series on which to operate, but you can use the following name-value arguments to select variables.
ResponseVariables
specifies the response series names in the input data from which residuals are inferred.PredictorVariables
specifies the predictor series names in the input data for a model regression component.Presample
specifies the input table or timetable of presample response data.PresampleResponseVariables
specifies the response series names fromPresample
.
See Also
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