forecast
Forecast vector error-correction (VEC) model responses
Syntax
Description
Conditional and Unconditional Forecasts for Numeric Arrays
returns a numeric array containing paths of minimum mean squared error (MMSE)
multivariate response forecasts Y
= forecast(Mdl
,numperiods
,Y0
)Y
over a length
numperiods
forecast horizon, using the fully specified
VEC(p – 1) model Mdl
. The forecasted
responses represent the continuation of the presample data in the numeric array
Y0
.
uses additional options specified by one or more name-value arguments.
Y
= forecast(Mdl
,numperiods
,Y0
,Name=Value
)forecast
returns numeric arrays when all optional
input data are numeric arrays. For example,
forecast(Mdl,10,Y0,X=Exo)
returns a numeric array
containing a 10-period forecasted response path from Mdl
and the numeric matrix of presample response data Y0
, and
specifies the numeric matrix of future predictor data for the model regression
component in the forecast horizon Exo
.
To produce a conditional forecast, specify future response data in a numeric
array by using the YF
name-value argument.
Unconditional Forecasts for Tables and Timetables
returns the table or timetable Tbl2
= forecast(Mdl
,numperiods
,Tbl1
)Tbl2
containing the length
numperiods
paths of multivariate MMSE response variable
forecasts, which result from computing unconditional forecasts from the VEC
model Mdl
. forecast
uses the table
or timetable of presample data Tbl1
to initialize the
response series. (since R2022b)
forecast
selects the variables in
Mdl.SeriesNames
to forecast, or it selects all variables
in Tbl1
. To select different response variables in
Tbl1
to forecast, use the
PresampleResponseVariables
name-value argument.
uses additional options specified by one or more name-value arguments. For
example, Tbl2
= forecast(Mdl
,numperiods
,Tbl1
,Name=Value
)forecast(Mdl,10,Tbl1,PresampleResponseVariables=["GDP"
"CPI"])
returns a timetable of response variables containing their
unconditional forecasts from the VEC model Mdl
, initialized
by the data in the GDP
and CPI
variables
of the timetable of presample data in Tbl1
. (since R2022b)
Conditional Forecasts for Tables and Timetables
returns the table or timetable Tbl2
= forecast(Mdl
,numperiods
,Tbl1
,InSample=InSample
,ResponseVariables=ResponseVariables
)Tbl2
containing the length
numperiods
paths of multivariate MMSE response variable
forecasts and corresponding forecast MSEs, which result from computing
conditional forecasts from the VEC model Mdl
.
forecast
uses the table or timetable of presample
data Tbl1
to initialize the response series.
InSample
is a table or timetable of future data in the
forecast horizon that forecast
uses to compute
conditional forecasts and ResponseVariables
specifies the
response variables in InSample
. (since R2022b)
uses additional options specified by one or more name-value arguments. (since R2022b)Tbl2
= forecast(Mdl
,numperiods
,Tbl1
,InSample=InSample
,ResponseVariables=ResponseVariables
,Name=Value
)
Examples
Return Matrix of VEC Model Forecasts
Consider a VEC model for the following seven macroeconomic series. Then, fit the model to the data and forecast responses 12 quarters into the future. Supply all required data in numeric matrices.
Gross domestic product (GDP)
GDP implicit price deflator
Paid compensation of employees
Nonfarm business sector hours of all persons
Effective federal funds rate
Personal consumption expenditures
Gross private domestic investment
Suppose that a cointegrating rank of 4 and one short-run term are appropriate, that is, consider a VEC(1) model.
Load the Data_USEconVECModel
data set.
load Data_USEconVECModel
For more information on the data set and variables, enter Description
at the command line.
Determine whether the data needs to be preprocessed by plotting the series on separate plots.
figure tiledlayout(2,2) nexttile plot(FRED.Time,FRED.GDP) title("Gross Domestic Product") ylabel("Index") xlabel("Date") nexttile plot(FRED.Time,FRED.GDPDEF) title("GDP Deflator") ylabel("Index") xlabel("Date") nexttile plot(FRED.Time,FRED.COE) title("Paid Compensation of Employees") ylabel("Billions of $") xlabel("Date") nexttile plot(FRED.Time,FRED.HOANBS) title("Nonfarm Business Sector Hours") ylabel("Index") xlabel("Date")
figure tiledlayout(2,2) nexttile plot(FRED.Time,FRED.FEDFUNDS) title("Federal Funds Rate") ylabel("Percent") xlabel("Date") nexttile plot(FRED.Time,FRED.PCEC) title("Consumption Expenditures") ylabel("Billions of $") xlabel("Date") nexttile plot(FRED.Time,FRED.GPDI) title("Gross Private Domestic Investment") ylabel("Billions of $") xlabel("Date")
Stabilize all series, except the federal funds rate, by applying the log transform. Scale the resulting series by 100 so that all series are on the same scale.
FRED.GDP = 100*log(FRED.GDP); FRED.GDPDEF = 100*log(FRED.GDPDEF); FRED.COE = 100*log(FRED.COE); FRED.HOANBS = 100*log(FRED.HOANBS); FRED.PCEC = 100*log(FRED.PCEC); FRED.GPDI = 100*log(FRED.GPDI);
Create a VEC(1) model using the shorthand syntax. Specify the variable names.
Mdl = vecm(7,4,1); Mdl.SeriesNames = FRED.Properties.VariableNames;
Mdl
is a vecm
model object. All properties containing NaN
values correspond to parameters to be estimated given data.
Estimate the model using the entire data set and the default options.
EstMdl = estimate(Mdl,FRED.Variables)
EstMdl = vecm with properties: Description: "7-Dimensional Rank = 4 VEC(1) Model" SeriesNames: "GDP" "GDPDEF" "COE" ... and 4 more NumSeries: 7 Rank: 4 P: 2 Constant: [14.1329 8.77841 -7.20359 ... and 4 more]' Adjustment: [7×4 matrix] Cointegration: [7×4 matrix] Impact: [7×7 matrix] CointegrationConstant: [-28.6082 -109.555 77.0912 ... and 1 more]' CointegrationTrend: [4×1 vector of zeros] ShortRun: {7×7 matrix} at lag [1] Trend: [7×1 vector of zeros] Beta: [7×0 matrix] Covariance: [7×7 matrix]
EstMdl
is an estimated vecm
model object. It is fully specified because all parameters have known values. By default, estimate
imposes the constraints of the H1 Johansen VEC model form by removing the cointegrating trend and linear trend terms from the model. Parameter exclusion from estimation is equivalent to imposing equality constraints to zero.
Forecast responses from the estimated model over a three-year horizon. Specify the entire data set as presample observations.
numperiods = 12; Y0 = FRED.Variables; Y = forecast(EstMdl,numperiods,Y0);
Y
is a 12-by-7 matrix of forecasted responses. Rows correspond to the forecast horizon, and columns correspond to the variables in EstMdl.SeriesNames
.
Plot the forecasted responses and the last 50 true responses.
fh = dateshift(FRED.Time(end),"end","quarter",1:12); figure; tiledlayout(2,2) nexttile h1 = plot(FRED.Time((end-49):end),FRED.GDP((end-49):end)); hold on h2 = plot(fh,Y(:,1)); title("Gross Domestic Product"); ylabel("Index (scaled)"); xlabel("Date"); h = gca; fill([FRED.Time(end) fh([end end]) FRED.Time(end)],h.YLim([1 1 2 2]),"k", ... FaceAlpha=0.1,EdgeColor="none"); legend([h1 h2],"True","Forecast",Location="best") hold off nexttile h1 = plot(FRED.Time((end-49):end),FRED.GDPDEF((end-49):end)); hold on h2 = plot(fh,Y(:,2)); title("GDP Deflator"); ylabel("Index (scaled)"); xlabel("Date"); h = gca; fill([FRED.Time(end) fh([end end]) FRED.Time(end)],h.YLim([1 1 2 2]),"k", ... FaceAlpha=0.1,EdgeColor="none"); legend([h1 h2],"True","Forecast",Location="best") hold off nexttile h1 = plot(FRED.Time((end-49):end),FRED.COE((end-49):end)); hold on h2 = plot(fh,Y(:,3)); title("Paid Compensation of Employees"); ylabel("Billions of $ (scaled)"); xlabel("Date"); h = gca; fill([FRED.Time(end) fh([end end]) FRED.Time(end)],h.YLim([1 1 2 2]),"k", ... FaceAlpha=0.1,EdgeColor="none"); legend([h1 h2],"True","Forecast",Location="best") hold off nexttile h1 = plot(FRED.Time((end-49):end),FRED.HOANBS((end-49):end)); hold on h2 = plot(fh,Y(:,4)); title("Nonfarm Business Sector Hours"); ylabel("Index (scaled)"); xlabel("Date"); h = gca; fill([FRED.Time(end) fh([end end]) FRED.Time(end)],h.YLim([1 1 2 2]),"k", ... FaceAlpha=0.1,EdgeColor="none"); legend([h1 h2],"True","Forecast",Location="best") hold off
figure tiledlayout(2,2) nexttile h1 = plot(FRED.Time((end-49):end),FRED.FEDFUNDS((end-49):end)); hold on h2 = plot(fh,Y(:,5)); title("Federal Funds Rate"); ylabel("Percent"); xlabel("Date"); h = gca; fill([FRED.Time(end) fh([end end]) FRED.Time(end)],h.YLim([1 1 2 2]),"k", ... FaceAlpha=0.1,EdgeColor="none"); legend([h1 h2],"True","Forecast",Location="best") hold off nexttile h1 = plot(FRED.Time((end-49):end),FRED.PCEC((end-49):end)); hold on h2 = plot(fh,Y(:,6)); title("Consumption Expenditures"); ylabel("Billions of $ (scaled)"); xlabel("Date"); h = gca; fill([FRED.Time(end) fh([end end]) FRED.Time(end)],h.YLim([1 1 2 2]),"k", ... FaceAlpha=0.1,EdgeColor="none"); legend([h1 h2],"True","Forecast",Location="best") hold off nexttile h1 = plot(FRED.Time((end-49):end),FRED.GPDI((end-49):end)); hold on h2 = plot(fh,Y(:,7)); title("Gross Private Domestic Investment"); ylabel("Billions of $ (scaled)"); xlabel("Date"); h = gca; fill([FRED.Time(end) fh([end end]) FRED.Time(end)],h.YLim([1 1 2 2]),"k", ... FaceAlpha=0.1,EdgeColor="none"); legend([h1 h2],"True","Forecast",Location="best") hold off
Compute Conditional Forecasts From Numeric Matrix of Future Response Data
This example is based on Return Matrix of VEC Model Forecasts. Forecast all response variables of the VEC model into a 3-year forecast horizon beyond the sampling data, given that the effective federal funds rate FEDFUNDS
is 0.5% during each future quarter.
Load the Data_USEconVECModel
data set.
load Data_USEconVECModel
Stabilize all series, except the federal funds rate, by applying the log transform. Scale the resulting series by 100 so that all series are on the same scale.
FRED.GDP = 100*log(FRED.GDP); FRED.GDPDEF = 100*log(FRED.GDPDEF); FRED.COE = 100*log(FRED.COE); FRED.HOANBS = 100*log(FRED.HOANBS); FRED.PCEC = 100*log(FRED.PCEC); FRED.GPDI = 100*log(FRED.GPDI);
Create a VEC(1) model using the shorthand syntax. Specify the variable names.
Mdl = vecm(7,4,1); Mdl.SeriesNames = FRED.Properties.VariableNames;
Estimate the model using the entire data set and the default options.
EstMdl = estimate(Mdl,FRED.Variables);
Suppose economists hypothesize that the effective federal funds rate will be at 0.5% for the next 12 quarters.
Create a matrix with the following qualities:
The matrix has 12 rows representing periods in the forecast horizon.
All columns associated with variables of
FRED
, except forFEDFUNDS
, are composed ofNaN
values.The column corresponding to the variable
FEDFUNDS
is composed of 0.5.
numperiods = 12;
CondF = NaN(numperiods,EstMdl.NumSeries);
idxFF = string(EstMdl.SeriesNames) == "FEDFUNDS";
CondF(:,idxFF) = 0.5*ones(numperiods,1);
CondF
is a 12-by-7 matrix of NaN
values, except for the column associated with FEDFUNDS
, which is a vector composed of the value 0.5. For each period in the forecast horizon, forecast
fills the NaN
elements of the matrix with forecasts, given the values of FEDFUNDS
.
Forecast all variables given the hypothesis by supplying the conditioning data CondF
. Supply the estimation sample as a presample to initialize the model.
Y = forecast(EstMdl,numperiods,FRED.Variables,YF=CondF);
Y
is a 12-by-7 matrix of forecasts and the fixed values in the column corresponding to FEDFUNDS
.
Plot the forecasts with the last few periods of the estimation sample.
fh = dateshift(FRED.Time(end),"end","quarter",1:numperiods); idx = find(~idxFF); figure; ht = tiledlayout(2,2); for j = idx(1:4) nexttile h1 = plot(FRED.Time((end-49):end),FRED{(end-49):end,j}); hold on h2 = plot(fh,Y(:,j)); title(EstMdl.SeriesNames(j)); xlabel("Date"); h = gca; fill([FRED.Time(end) fh([end end]) FRED.Time(end)],h.YLim([1 1 2 2]),"k", ... FaceAlpha=0.1,EdgeColor="none"); legend([h1 h2],"True","Forecast",Location="best") hold off end title(ht,"Forecasts With FEDFUNDS = 0.5")
figure; ht = tiledlayout(2,1); for j = idx(5:6) nexttile h1 = plot(FRED.Time((end-49):end),FRED{(end-49):end,j}); hold on h2 = plot(fh,Y(:,j)); title(EstMdl.SeriesNames(j)); xlabel("Date"); h = gca; fill([FRED.Time(end) fh([end end]) FRED.Time(end)],h.YLim([1 1 2 2]),"k", ... FaceAlpha=0.1,EdgeColor="none"); legend([h1 h2],"True","Forecast",Location="best") hold off end title(ht,"Forecasts With FEDFUNDS = 0.5")
Estimate Forecast Intervals
Analyze forecast accuracy using forecast intervals over a three-year horizon. This example follows from Return Matrix of VEC Model Forecasts.
Load the Data_USEconVECModel
data set and preprocess the data.
load Data_USEconVECModel
FRED.GDP = 100*log(FRED.GDP);
FRED.GDPDEF = 100*log(FRED.GDPDEF);
FRED.COE = 100*log(FRED.COE);
FRED.HOANBS = 100*log(FRED.HOANBS);
FRED.PCEC = 100*log(FRED.PCEC);
FRED.GPDI = 100*log(FRED.GPDI);
Estimate a VEC(1) model. Reserve the last three years of data to assess forecast accuracy. Assume that the appropriate cointegration rank is 4, and the H1 Johansen form is appropriate for the model.
bfh = FRED.Time(end) - years(3); estIdx = FRED.Time < bfh; Mdl = vecm(7,4,1); Mdl.SeriesNames = FRED.Properties.VariableNames; EstMdl = estimate(Mdl,FRED{estIdx,:});
Forecast responses from the estimated model over a three-year horizon. Specify all in-sample observations as a presample. Return the MSE of the forecasts.
numperiods = 12; Y0 = FRED{estIdx,:}; [Y,YMSE] = forecast(EstMdl,numperiods,Y0);
Y
is a 12-by-7 matrix of forecasted responses. YMSE
is a 12-by-1 cell vector of 7-by-7 matrices corresponding to the MSEs.
Extract the main diagonal elements from the matrices in each cell of YMSE
. Apply the square root of the result to obtain standard errors.
extractMSE = @(x)diag(x)'; MSE = cellfun(extractMSE,YMSE,UniformOutput=false); SE = sqrt(cell2mat(MSE));
Estimate approximate 95% forecast intervals for each response series.
YFI = zeros(numperiods,Mdl.NumSeries,2); YFI(:,:,1) = Y - 2*SE; YFI(:,:,2) = Y + 2*SE;
Plot the forecasted responses and the last 40 true responses.
figure ht = tiledlayout(2,2); for j = 1:4 nexttile h1 = plot(FRED.Time((end-39):end),FRED{(end-39):end,j}); hold on h2 = plot(FRED.Time(~estIdx),Y(:,j)); h3 = plot(FRED.Time(~estIdx),YFI(:,j,1),"k--"); plot(FRED.Time(~estIdx),YFI(:,j,2),"k--"); title(EstMdl.SeriesNames(j)); xlabel("Date"); h = gca; fill([bfh h.XLim([2 2]) bfh],h.YLim([1 1 2 2]),"k", ... FaceAlpha=0.1,EdgeColor="none"); legend([h1 h2 h3],"Observed","Forecast","Forecast interval", ... Location="best"); hold off end title(ht,"Forecasts and 95% Forecast Intervals")
figure ht = tiledlayout(2,2); for j = 5:7 nexttile h1 = plot(FRED.Time((end-39):end),FRED{(end-39):end,j}); hold on h2 = plot(FRED.Time(~estIdx),Y(:,j)); h3 = plot(FRED.Time(~estIdx),YFI(:,j,1),"k--"); plot(FRED.Time(~estIdx),YFI(:,j,2),"k--"); title(EstMdl.SeriesNames(j)); xlabel("Date"); h = gca; fill([bfh h.XLim([2 2]) bfh],h.YLim([1 1 2 2]),"k", ... FaceAlpha=0.1,EdgeColor="none"); legend([h1 h2 h3],"Observed","Forecast","Forecast interval", ... Location="best"); hold off end title(ht,"Forecasts and 95% Forecast Intervals")
Return Timetable of Forecasts and Array of Forecast MSEs
Since R2022b
Consider a VEC model for the following seven macroeconomic series, and then fit the model to a timetable of response data. This example is based on Return Matrix of VEC Model Forecasts.
Load and Preprocess Data
Load the Data_USEconVECModel
data set.
load Data_USEconVECModel
DTT = FRED;
DTT.GDP = 100*log(DTT.GDP);
DTT.GDPDEF = 100*log(DTT.GDPDEF);
DTT.COE = 100*log(DTT.COE);
DTT.HOANBS = 100*log(DTT.HOANBS);
DTT.PCEC = 100*log(DTT.PCEC);
DTT.GPDI = 100*log(DTT.GPDI);
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.
DTT = rmmissing(DTT); T = height(DTT)
T = 240
DTT
does not contain any missing values.
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;
DTT
is regular with respect to time.
Create Model Template for Estimation
Create a VEC(1) model by using the shorthand syntax. Specify the variable names.
Mdl = vecm(7,4,1); Mdl.SeriesNames = DTT.Properties.VariableNames;
Mdl
is a vecm
model object. All properties containing NaN
values correspond to parameters to be estimated given data.
Fit Model to Data
Estimate the model by supplying the timetable of data DTT
. By default, because the number of variables in Mdl.SeriesNames
is the number of variables in DTT
, estimate
fits the model to all the variables in DTT
.
EstMdl = estimate(Mdl,DTT);
EstMdl
is an estimated vecm
model object.
Forecast Responses and Compute Forecast MSEs
Forecast responses from the estimated model over a three-year horizon. Specify the entire data set DTT
as a presample observations.
numperiods = 12; [Tbl,YMSE] = forecast(EstMdl,numperiods,DTT); size(Tbl)
ans = 1×2
12 7
tail(DTT)
Time GDP GDPDEF COE HOANBS FEDFUNDS PCEC GPDI ___________ ______ ______ ______ ______ ________ ______ ______ 01-Jan-2015 978.6 469.42 915.93 470.1 0.11 940.09 802.11 01-Apr-2015 979.8 469.97 917.34 470.57 0.13 941.25 802.29 01-Jul-2015 980.6 470.28 918.4 470.52 0.14 942.2 803.01 01-Oct-2015 981.04 470.51 919.95 471.33 0.24 942.86 802.61 01-Jan-2016 981.37 470.62 919.95 471.67 0.36 943.33 801.86 01-Apr-2016 982.28 471.19 921.5 472.09 0.38 944.88 800.22 01-Jul-2016 983.5 471.54 922.78 472.24 0.4 945.97 801.21 01-Oct-2016 984.48 472.06 923.69 472.47 0.54 947.12 804.13
head(Tbl)
Time GDP_Responses GDPDEF_Responses COE_Responses HOANBS_Responses FEDFUNDS_Responses PCEC_Responses GPDI_Responses ___________ _____________ ________________ _____________ ________________ __________________ ______________ ______________ 01-Jan-2017 985.7 472.53 924.74 472.87 0.3725 948.18 806.74 01-Apr-2017 986.82 472.93 925.75 473.21 0.33795 949.24 808.66 01-Jul-2017 987.92 473.31 926.78 473.57 0.30002 950.29 810.45 01-Oct-2017 988.99 473.67 927.82 473.94 0.27518 951.35 812.12 01-Jan-2018 990.07 474.02 928.88 474.33 0.263 952.42 813.74 01-Apr-2018 991.14 474.37 929.95 474.74 0.26045 953.49 815.32 01-Jul-2018 992.22 474.71 931.04 475.15 0.26472 954.56 816.86 01-Oct-2018 993.29 475.05 932.14 475.56 0.27283 955.64 818.35
YMSE
YMSE=12×1 cell array
{7x7 double}
{7x7 double}
{7x7 double}
{7x7 double}
{7x7 double}
{7x7 double}
{7x7 double}
{7x7 double}
{7x7 double}
{7x7 double}
{7x7 double}
{7x7 double}
YMSE{6}
ans = 7×7
7.6245 1.6879 7.7978 6.3846 3.5735 5.2342 26.8879
1.6879 1.9506 1.7640 0.4391 1.6560 1.2281 4.4627
7.7978 1.7640 8.8184 6.9137 3.6937 5.4552 28.3538
6.3846 0.4391 6.9137 7.4894 2.9271 4.2783 25.3822
3.5735 1.6560 3.6937 2.9271 4.3945 2.1872 12.6306
5.2342 1.2281 5.4552 4.2783 2.1872 4.1945 18.0819
26.8879 4.4627 28.3538 25.3822 12.6306 18.0819 113.1428
Tbl
is a 12-by-7 matrix of forecasted responses (denoted responseVariable
_Responses
). The timestamps of Tbl
follow directly from the timestamps of DTT
, and they have the same sampling frequency. YMSE is a 12-by-1 cell array of 7-by-7 forecast MSE matrices. For example, the forecast covariance of GDP
and COE
in period 6 of the forecast horizon if element (1,3) of the matrix in YMSE{6}
, which is 7.7978.
Forecast VECX Model
Since R2022b
Consider the model and data in Return Matrix of VEC Model Forecasts.
Load Data
Load the Data_USEconVECModel
data set.
load Data_USEconVECModel
The Data_Recessions
data set contains the beginning and ending serial dates of recessions. Load this data set. Convert the matrix of date serial numbers to a datetime array.
load Data_Recessions dtrec = datetime(Recessions,ConvertFrom="datenum");
Preprocess Data
Remove the exponential trend from the series, and then scale them by a factor of 100.
DTT = FRED; DTT.GDP = 100*log(DTT.GDP); DTT.GDPDEF = 100*log(DTT.GDPDEF); DTT.COE = 100*log(DTT.COE); DTT.HOANBS = 100*log(DTT.HOANBS); DTT.PCEC = 100*log(DTT.PCEC); DTT.GPDI = 100*log(DTT.GPDI);
Create a dummy variable that identifies periods in which the U.S. was in a recession or worse. Specifically, the variable should be 1
if FRED.Time
occurs during a recession, and 0
otherwise. Include the variable with the FRED
data.
isin = @(x)(any(dtrec(:,1) <= x & x <= dtrec(:,2))); DTT.IsRecession = double(arrayfun(isin,DTT.Time));
Prepare Timetable for Estimation
Remove all missing values from the table.
DTT = rmmissing(DTT);
To make the series regular, shift all dates to the first day of the quarter.
dt = DTT.Time; dt = dateshift(dt,"start","quarter"); DTT.Time = dt;
DTT
is regular with respect to time.
Create Model Template for Estimation
Create a VEC(1) model using the shorthand syntax. Assume that the appropriate cointegration rank is 4. You do not have to specify the presence of a regression component when creating the model. Specify the variable names.
Mdl = vecm(7,4,1); Mdl.SeriesNames = DTT.Properties.VariableNames(1:end-1);
Fit Model to Data
Estimate the model using all but the last three years of data. Specify the predictor identifying whether the observation was measured during a recession.
bfh = DTT.Time(end) - years(3);
fh = DTT.Time(DTT.Time >= bfh);
EstSample = DTT(DTT.Time < bfh,:);
FSample = DTT(fh,:);
EstMdl = estimate(Mdl,EstSample,PredictorVariables="IsRecession");
Forecast Responses
Forecast a path of quarterly responses three years into the future.
numperiods = numel(fh); Tbl = forecast(EstMdl,numperiods,EstSample, ... InSample=FSample,PredictorVariables="IsRecession"); head(Tbl(:,endsWith(Tbl.Properties.VariableNames,"_Responses")))
Time GDP_Responses GDPDEF_Responses COE_Responses HOANBS_Responses FEDFUNDS_Responses PCEC_Responses GPDI_Responses ___________ _____________ ________________ _____________ ________________ __________________ ______________ ______________ 01-Jan-2014 974.87 468.25 911.21 467.31 0.47511 936.25 793.63 01-Apr-2014 975.81 468.6 912.19 467.82 0.63807 937.22 794.68 01-Jul-2014 976.67 468.91 913.19 468.3 0.72011 938.16 795.47 01-Oct-2014 977.53 469.21 914.16 468.77 0.76135 939.08 796.33 01-Jan-2015 978.38 469.49 915.12 469.2 0.7691 939.98 797.17 01-Apr-2015 979.22 469.77 916.06 469.62 0.75747 940.86 798 01-Jul-2015 980.05 470.04 916.99 470.02 0.73223 941.74 798.83 01-Oct-2015 980.89 470.31 917.91 470.41 0.69828 942.62 799.67
Tbl
is a 12-by-15 matrix of variables in FSample
and forecasted responses (variables named responseVariable
_Responses
, for each response responseVariable
in the model).
Plot the forecasted responses and the last 50 true responses.
figure; tiledlayout(2,2) for j = EstMdl.SeriesNames(1:4) nexttile h1 = plot(DTT.Time((end-49):end),DTT{(end-49):end,j}); hold on h2 = plot(Tbl.Time,Tbl{:,j+"_Responses"}); title(j); xlabel("Date"); h = gca; fill([DTT.Time(end) bfh([end end]) DTT.Time(end)],h.YLim([1 1 2 2]),"k", ... FaceAlpha=0.1,EdgeColor="none"); legend([h1 h2],"True","Forecast",Location="best") hold off end
figure tiledlayout(2,2) for j = EstMdl.SeriesNames(5:7) nexttile h1 = plot(DTT.Time((end-49):end),DTT{(end-49):end,j}); hold on h2 = plot(Tbl.Time,Tbl{:,j+"_Responses"}); title(j); xlabel("Date"); h = gca; fill([DTT.Time(end) bfh([end end]) DTT.Time(end)],h.YLim([1 1 2 2]),"k", ... FaceAlpha=0.1,EdgeColor="none"); legend([h1 h2],"True","Forecast",Location="best") hold off end
Return Timetable of Conditional Forecasts
Since R2022b
This example is based on Return Timetable of Forecasts and Array of Forecast MSEs. Forecast all response variables of the VEC model into a 3-year forecast horizon beyond the sampling data, given that the effective federal funds rate FEDFUNDS
is 0.5% during each future quarter.
Load and Preprocess Data
Load the Data_USEconVECModel
data set.
load Data_USEconVECModel
DTT = FRED;
DTT.GDP = 100*log(DTT.GDP);
DTT.GDPDEF = 100*log(DTT.GDPDEF);
DTT.COE = 100*log(DTT.COE);
DTT.HOANBS = 100*log(DTT.HOANBS);
DTT.PCEC = 100*log(DTT.PCEC);
DTT.GPDI = 100*log(DTT.GPDI);
Prepare Timetable for Estimation
Remove all missing values from the table.
DTT = rmmissing(DTT);
To make the series regular, shift all dates to the first day of the quarter.
dt = DTT.Time; dt = dateshift(dt,"start","quarter"); DTT.Time = dt;
DTT
is regular with respect to time.
Create Model Template for Estimation
Create a VEC(1) model using the shorthand syntax. Specify the variable names.
Mdl = vecm(7,4,1); Mdl.SeriesNames = DTT.Properties.VariableNames;
Mdl
is a vecm
model object. All properties containing NaN
values correspond to parameters to be estimated given data.
Fit Model to Data
Estimate the model. Pass the entire timetable DTT
.
EstMdl = estimate(Mdl,DTT);
Prepare for Conditional Forecast of Estimated Model
Suppose economists hypothesize that the effective federal funds rate will be at 0.5% for the next 12 quarters.
Create a timetable with the following qualities:
The timestamps are regular with respect to the estimation sample timestamps and they are ordered from Q1 of 2017 through Q4 of 2019.
All variables of DTT, except for
FEDFUNDS
, are a 12-by-1 vector ofNaN
values.FEDFUNDS
is a 12-by-1 vector, where each element is 0.5.
numperiods = 12;
shdt = DTT.Time(end) + calquarters(1:numperiods);
DTTCondF = retime(DTT,shdt,"fillwithmissing");
DTTCondF.FEDFUNDS = 0.5*ones(numperiods,1);
DTTCondF
is a 12-by-7 timetable that follows directly, in time, from DTT
, and both timetables have the same variables. All variables in DTTCondF
contain NaN
values, except for FEDFUNDS
, which is a vector composed of the value 0.5.
Perform Conditional Simulation of Estimated Model
Forecast all response variables, given the hypothesis, by supplying the conditioning data DTTCondF
and specifying the response variable names. Supply the estimation sample as a presample to initialize the model.
Tbl = forecast(EstMdl,numperiods,DTT, ...
InSample=DTTCondF,ResponseVariables=EstMdl.SeriesNames);
size(Tbl)
ans = 1×2
12 14
idx = endsWith(Tbl.Properties.VariableNames,"_Responses");
head(Tbl(:,idx))
Time GDP_Responses GDPDEF_Responses COE_Responses HOANBS_Responses FEDFUNDS_Responses PCEC_Responses GPDI_Responses ___________ _____________ ________________ _____________ ________________ __________________ ______________ ______________ 01-Jan-2017 985.73 472.53 924.76 472.89 0.5 948.2 806.83 01-Apr-2017 986.89 472.96 925.8 473.27 0.5 949.27 808.96 01-Jul-2017 988.01 473.36 926.87 473.65 0.5 950.34 810.86 01-Oct-2017 989.12 473.74 927.94 474.04 0.5 951.42 812.62 01-Jan-2018 990.22 474.12 929.04 474.45 0.5 952.5 814.28 01-Apr-2018 991.31 474.49 930.14 474.85 0.5 953.59 815.85 01-Jul-2018 992.39 474.86 931.25 475.25 0.5 954.67 817.35 01-Oct-2018 993.47 475.24 932.36 475.65 0.5 955.76 818.79
Tbl
is a 12-by-14 matrix of forecasts of all response variables of the VEC model in the forecast horizon, given FEDFUNDS
is 0.5%. GDP_Responses
contains the forecasts of the transformed GDP series. FEDFUNDS_Responses
is a 12-by-1 vector composed of the value 0.5.
Return Multiple Conditional Forecast Paths
Since R2022b
This example is based on Return Timetable of Forecasts and Array of Forecast MSEs. Forecast all response variables of the VEC model into a 1-year forecast horizon beyond the sampling data, given several hypotheses economists make on the effective federal funds rate FEDFUNDS
during each quarter of the next year after the sampling period.
Load the Data_USEconVECModel
data set.
load Data_USEconVECModel
DTT = FRED;
DTT.GDP = 100*log(DTT.GDP);
DTT.GDPDEF = 100*log(DTT.GDPDEF);
DTT.COE = 100*log(DTT.COE);
DTT.HOANBS = 100*log(DTT.HOANBS);
DTT.PCEC = 100*log(DTT.PCEC);
DTT.GPDI = 100*log(DTT.GPDI);
Remove all missing values from the table.
DTT = rmmissing(DTT);
To make the series regular, shift all dates to the first day of the quarter.
dt = DTT.Time; dt = dateshift(dt,"start","quarter"); DTT.Time = dt;
DTT
is regular with respect to time.
Create a VEC(1) model using the shorthand syntax. Specify the variable names.
Mdl = vecm(7,4,1); Mdl.SeriesNames = DTT.Properties.VariableNames;
Estimate the model. Pass the entire timetable DTT
.
EstMdl = estimate(Mdl,DTT);
Assuming the effective federal funds rate is 0.1%, 0.25%, 0.5%, 0.75%, and 1% percent throughout a 1-year forecast horizon, generate a forecast path for all response variables under each scenario.
Create a timetable with the following qualities:
The timestamps are regular with respect to the estimation sample timestamps and they are ordered from Q1 of 2017 through Q4 of 2017.
The variable
FEDFUNDS
is a 4-by-5 matrix, where each column is composed of each of the assumptions on the value of the effective federal funds rate in the forecast horizon; the elements of the first column are 0.1, elements of the second column are 0.25, and so on.Each other response variable is a 4-by-5 matrix of
NaN
values to be filled with forecasted paths byforecast
.
numperiods = 4;
shdt = DTT.Time(end) + calquarters(1:numperiods);
DTTCondF = retime(DTT,shdt,"fillwithmissing");
DTTCondF = varfun(@(x)nan(numperiods,5),DTTCondF);
DTTCondF.Properties.VariableNames = EstMdl.SeriesNames;
DTTCondF.FEDFUNDS = ones(numperiods,1)*[0.1 0.25 0.5 0.75 1];
DTTCondF
DTTCondF=4×7 timetable
Time GDP GDPDEF COE HOANBS FEDFUNDS PCEC GPDI
___________ _______________________________ _______________________________ _______________________________ _______________________________ ___________________________________ _______________________________ _______________________________
01-Jan-2017 NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN 0.1 0.25 0.5 0.75 1 NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN
01-Apr-2017 NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN 0.1 0.25 0.5 0.75 1 NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN
01-Jul-2017 NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN 0.1 0.25 0.5 0.75 1 NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN
01-Oct-2017 NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN 0.1 0.25 0.5 0.75 1 NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN
DTTCondF
is a 4-by-7 timetable that follows directly, in time, from DTT
, and both timetables have the same variables. Each variable in DTTCondF
contains a 4-by-5 matrix of NaN
values, except for FEDFUNDS
, which is a matrix with each column containing a different scenario for the conditional forecasts.
Forecast all response variables, given the hypotheses, by supplying the conditioning data DTTCondF
and specifying the response variable names. Supply the estimation sample as a presample to initialize the model. Return the forecast MSE matrices.
[Tbl,YMSE] = forecast(EstMdl,numperiods,DTT, ...
InSample=DTTCondF,ResponseVariables=EstMdl.SeriesNames);
size(Tbl)
ans = 1×2
4 14
idx = endsWith(Tbl.Properties.VariableNames,"_Responses");
head(Tbl(:,idx))
Time GDP_Responses GDPDEF_Responses COE_Responses HOANBS_Responses FEDFUNDS_Responses PCEC_Responses GPDI_Responses ___________ ______________________________________________ ______________________________________________ ______________________________________________ ______________________________________________ ___________________________________ ______________________________________________ ______________________________________________ 01-Jan-2017 985.65 985.68 985.73 985.77 985.82 472.51 472.52 472.53 472.54 472.55 924.7 924.72 924.76 924.79 924.82 472.83 472.85 472.89 472.94 472.98 0.1 0.25 0.5 0.75 1 948.14 948.16 948.2 948.23 948.27 806.54 806.65 806.83 807.01 807.2 01-Apr-2017 986.73 986.79 986.89 986.98 987.08 472.9 472.92 472.96 472.99 473.03 925.67 925.72 925.8 925.88 925.97 473.13 473.18 473.27 473.35 473.44 0.1 0.25 0.5 0.75 1 949.2 949.23 949.27 949.31 949.36 808.17 808.47 808.96 809.45 809.94 01-Jul-2017 987.83 987.9 988.01 988.12 988.24 473.26 473.29 473.36 473.42 473.48 926.69 926.76 926.87 926.97 927.08 473.5 473.55 473.65 473.74 473.84 0.1 0.25 0.5 0.75 1 950.26 950.29 950.34 950.4 950.45 810.06 810.36 810.86 811.36 811.86 01-Oct-2017 988.93 989 989.12 989.24 989.37 473.6 473.65 473.74 473.83 473.92 927.74 927.82 927.94 928.07 928.2 473.9 473.96 474.04 474.13 474.22 0.1 0.25 0.5 0.75 1 951.33 951.36 951.42 951.48 951.54 811.86 812.15 812.62 813.1 813.58
YMSE
YMSE=4×1 cell array
{7x7 double}
{7x7 double}
{7x7 double}
{7x7 double}
YMSE{4}
ans = 7×7
2.9103 0.2459 2.6926 2.2954 0 1.9785 10.5522
0.2459 0.6435 0.2598 -0.2005 0 0.2656 0.1772
2.6926 0.2598 3.1251 2.3680 0 1.9150 10.3987
2.2954 -0.2005 2.3680 3.0306 0 1.5138 10.0253
0 0 0 0 0 0 0
1.9785 0.2656 1.9150 1.5138 0 1.7880 6.7155
10.5522 0.1772 10.3987 10.0253 0 6.7155 50.7359
Tbl
is a 4-by-14 matrix of forecasts of all response variables of the VEC model in the forecast horizon, given each assumption on FEDFUNDS
. GDP_Responses
contains the matrix of 5 forecast paths of the transformed GDP series from matrix of 5 forecast paths. Each path uses the corresponding assumption about the value of FEDFUNDS_Responses
.
YMSE
is a 4-by-1 cell vector of 7-by-7 forecast MSE matrices for each period in the forecast horizon. The MSE matrices apply to each forecast path, and all elements of each matrix corresponding to the conditioning variable are 0.
Input Arguments
numperiods
— Forecast horizon
positive integer
Forecast horizon, or the number of time points in the forecast period, specified as a positive integer.
Data Types: double
Y0
— Presample response data
numeric matrix | numeric array
Presample response data that provides initial values for the forecasts, specified as a
numpreobs
-by-numseries
numeric matrix or a
numpreobs
-by-numseries
-by-numprepaths
numeric array. Use Y0
only when you supply optional data inputs as
numeric arrays.
numpreobs
is the number of presample observations.
numseries
is the number of response series
(Mdl.NumSeries
). 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, forecast
uses the latest
Mdl.P
observations only.
Each column corresponds to the response series name in
Mdl.SeriesNames
.
Pages correspond to separate, independent paths.
If you compute unconditional forecasts (that is, you do not specify the
YF
name-value argument),forecast
initializes each forecasted path (page) using the corresponding page ofY0
. Therefore, the output argumentY
hasnumpaths
=numprepaths
pages.If you compute conditional forecasts by specifying future response data in
YF
:forecast
takes one of these actions:If
Y0
is a matrix,forecast
initializes each response path (page) inYF
using the corresponding presample response inY0
. Therefore,numpaths
is the number of paths inYF
, and all paths in the output argumentY
derive from common initial conditions.If
YF
is a matrix,forecast
generatesnumprepaths
forecast paths, initialized by each presample response path inY0
, but the future response data, from which to condition the forecasts, is the same among all paths. Therefore,numprepaths
is the number of paths in the output argumentY
, and all paths evolve from possibly different initial conditions.Otherwise,
numpaths
is the minimum betweennumprepaths
and the number of pages inYF
, andforecast
appliesY0(:,:,
to initialize forecasting pathj
)
, forj
= 1,…,j
numpaths
.
Data Types: double
Tbl1
— Presample data
table | timetable
Since R2022b
Presample response data that provides initial values for the forecasts, specified as a
table or timetable with numprevars
variables and
numpreobs
rows. forecast
returns the
forecasted response variable in the output table or timetable Tbl2
,
which is commensurate with Tbl1
.
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,
forecast
uses the latest Mdl.P
observations only.
Each selected response variable is a
numpreobs
-by-numprepaths
numeric matrix. You
can optionally specify numseries
response variables by using the
PresampleResponseVariables
name-value argument.
Paths (columns) within a particular response variable are independent, but path
of all variables correspond, for
j
=
1,…,j
numprepaths
. The following conditions apply:
If you compute unconditional forecasts (that is, you do not specify the
InSample
andResponseVariables
name-value arguments),forecast
initializes each forecasted path per selected response variable using the corresponding path inTbl1
. Therefore, each forecasted response variable in the output argumentTbl2
is anumperiods
-by-numprepaths
matrix.If you compute conditional forecasts by specifying future response data in
InSample
and corresponding response variables from the data by usingResponseVariables
,forecast
takes one of these actions:If the selected presample response variables are vectors,
forecast
initializes each forecast path (column) of the selected response variables inInSample
by using the corresponding presample variable inTbl1
. Therefore, all paths in the forecasted response variables evolve from common initial conditions.If the selected response variables in
InSample
are vectors,forecast
generatesnumprepaths
forecast paths, initialized by the paths of each selected presample response variable inTbl1
, but the future response data, from which to condition the forecasts, is the same among all paths. Therefore,numpaths
=numprepaths
is the number of paths in all forecasted response variables, and all paths evolve from possibly different initial conditions.Otherwise,
numpaths
is the minimum betweennumprepaths
and the number of paths in each selected response variable inInSample
. For each selected presample and future sample response variable
and each pathResponseK
= 1,…,j
numpaths
,forecast
appliesTbl1.
to initialize the conditional forecast for the response data inResponseK
(:,j
)Tbl2.
ResponseK
(:,
.j
)
If Tbl1
is a timetable, all the following conditions must be true:
Tbl1
must represent a sample with a regular datetime time step (seeisregular
).The inputs
InSample
andTbl1
must be consistent in time such thatTbl1
immediately precedesInSample
with respect to the sampling frequency and order.The datetime vector of sample timestamps
Tbl1.Time
must be ascending or descending.
If Tbl1
is a table, the last row contains the latest presample
observation.
InSample
— Future time series response or predictor data
table | timetable
Since R2022b
Future time series response or predictor data, specified as a table or timetable. InSample
contains numvars
variables, including numseries
response variables yt or numpreds
predictor variables xt for the model regression component. You can specify InSample
only when you specify Tbl1
.
Use InSample
in the following situations:
Perform conditional simulation. You must also supply the response variable names to select response data in
InSample
by using theResponseVariables
name-value argument.Supply future predictor data for either unconditional or conditional simulation. To supply predictor data, you must specify predictor variable names in
InSample
by using thePredictorVariables
name-value argument. Otherwise,forecast
ignores the model regression component.
Each row corresponds to an observation in the forecast horizon, the first row is the earliest observation, and measurements in each row, among all paths, occur simultaneously. Specifically, row
of variable j
(VariableK
InSample.
) contains observations VariableK
(j
,:)
periods into the future, or the j
-period-ahead forecasts. j
InSample
must have at least numperiods
rows to cover the forecast horizon. If you supply more rows than necessary, forecast
uses only the first numperiods
rows.
Each selected response variable is a numeric matrix. For each selected response variable
, columns are separate, independent paths. Specifically, path K
of response variable j
captures the state, or knowledge, of ResponseK
as it evolves from the presample past (for example, ResponseK
Tbl1.
) into the future. For each selected response variable ResponseK
:ResponseK
If the selected presample response variables in
Tbl1
are vectors,forecast
initializes each forecast path (column) of the selected response variables inInSample
by using the corresponding presample variable inTbl1
. Therefore, all paths in the forecasted response variables of the outputTbl2
evolve from common initial conditions.If the selected response variables in
InSample
are vectors,forecast
generatesnumprepaths
forecast paths, initialized by the paths of each selected presample response variable inTbl1
, but the future response data, from which to condition the forecasts, is the same among all paths. Therefore,numpaths
=numprepaths
is the number of paths in all forecasted response variables, and all paths evolve from possibly different initial conditions.Otherwise,
numpaths
is the minimum betweennumprepaths
and the number of paths in each selected response variable inInSample
. For each selected presample and future sample response variable
and each pathResponseK
= 1,…,j
numpaths
,forecast
appliesTbl1.
to initialize the conditional forecast for the response data inResponseK
(:,j
)Tbl2.
ResponseK
(:,
.j
)
Each predictor variable is a numeric vector. All predictor variables are present in the regression component of each response equation and apply to all response paths.
If InSample
is a timetable, the following conditions apply:
InSample
must represent a sample with a regular datetime time step (seeisregular
).The datetime vector
InSample.Time
must be ascending or descending.Tbl1
must immediately precedeInSample
, with respect to the sampling frequency.
If InSample
is a table, the last row contains the latest
observation.
Elements of the response variables of InSample
can be numeric scalars or missing values (indicated by NaN
values). forecast
treats numeric scalars as deterministic future responses that are known in advance, for example, set by policy. forecast
forecasts responses for corresponding NaN
values conditional on the known values. Elements of selected predictor variables must be numeric scalars.
By default, forecast
computes conventional MMSE forecasts and forecast
MSEs without a regression component in the model (each selected response variable is a
numperiods
-by-numprepaths
matrix composed of
NaN
values indicating a complete lack of knowledge of the future
state of the responses in the forecast horizon).
For more details, see Algorithms.
Example: Consider forecasting one path from a model composed of two
response series, GDP
and CPI
, three
periods into the future. Suppose that you have prior knowledge about some of
the future values of the responses, and you want to forecast the unknown
responses conditional on your knowledge. Specify InSample
as a matrix containing the values that you know, and use
NaN
for values you do not know but want to forecast.
For example, InSample=array2table([2 NaN; 0.1 NaN; NaN
NaN],VariableNames=["GDP" "CPI"])
specifies that you have no
knowledge of the future values of CPI
, but you know that
GDP
is 2, 0.1, and unknown in periods 1, 2, and 3,
respectively, in the forecast horizon.
ResponseVariables
— Variables to select from InSample
to treat as response variables yt
string vector | cell vector of character vectors | vector of integers | logical vector
Since R2022b
Variables to select from InSample
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 inInSample.Properties.VariableNames
A length
numseries
vector of unique indices (integers) of variables to select fromInSample.Properties.VariableNames
A length
numvars
logical vector, whereResponseVariables(
selects variablej
) = true
fromj
InSample.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.
To compute conditional forecasts, you must specify
ResponseVariables
to select the response variables in
InSample
for the conditioning data.
ResponseVariables
applies only when you specify
InSample
.
By default, forecast
computes conventional MMSE forecasts and
forecast MSEs.
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
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: forecast(Mdl,10,Y0,X=Exo)
returns a numeric array
containing a 10-period forecasted response path from Mdl
and
the numeric matrix of presample response data Y0
, and specifies
the numeric matrix of future predictor data for the model regression component in
the forecast horizon Exo
.
PresampleResponseVariables
— Variables to select from Tbl1
to use for presample response data
string vector | cell vector of character vectors | vector of integers | logical vector
Since R2022b
Variables to select from Tbl1
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 inTbl1.Properties.VariableNames
A length
numseries
vector of unique indices (integers) of variables to select fromTbl1.Properties.VariableNames
A length
numprevars
logical vector, wherePresampleResponseVariables(
selects variablej
) = true
fromj
Tbl1.Properties.VariableNames
, andsum(PresampleResponseVariables)
isnumseries
The selected variables must be numeric vectors and cannot contain missing values
(NaN
).
PresampleResponseNames
does not need to contain the same names as
in Mdl.SeriesNames
; forecast
uses the data
in selected variable
PresampleResponseVariables(
as a
presample for j
)Mdl.SeriesNames(
.j
)
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: 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
— Forecasted time series of predictor data xt
numeric matrix
Forecasted time series of predictor data xt
to include in the model regression component, specified as a numeric matrix containing
numpreds
columns. Use X
only when you supply
Y0
.
numpreds
is the number of predictor variables
(size(Mdl.Beta,2)
).
Each row corresponds to an observation in the forecast horizon, and measurements in
each row occur simultaneously. Specifically, row
(j
X(
) contains the predictor
observations j
,:)
periods into the future, or
the j
-period-ahead forecasts.
j
X
must have at least numperiods
rows. If you
supply more rows than necessary, forecast
uses only the earliest
numperiods
observations. The first row contains the earliest
observation. forecast
does not use the regression component in
the presample period.
Each column is an individual predictor variable. All predictor variables are present in the regression component of each response equation.
forecast
applies X
to each path (page);
that is, X
represents one path of observed predictors.
To maintain model consistency into the forecast horizon, specify forecasted predictors
when Mdl
has a regression component.
By default, forecast
excludes the regression component,
regardless of its presence in Mdl
.
Data Types: double
YF
— Future multivariate response series
numeric matrix | numeric array
Future multivariate response series data for conditional forecasting, specified as a numeric
matrix or array containing numseries
columns. Use YF
only when you supply Y0
.
Each row corresponds to observations in the forecast horizon, and the first row is the
earliest observation. Specifically, row j
in sample path
k
(YF(
)
contains the responses j
,:,k
)j
periods into the future, or the
-period-ahead forecasts.
j
YF
must have at least numperiods
rows to cover
the forecast horizon. If you supply more rows than necessary,
forecast
uses only the first numperiods
rows.
Each column corresponds to the response variable name in
Mdl.SeriesNames
.
Each page corresponds to a separate sample path. Specifically, path
k
(YF(:,:,
)
captures the state, or knowledge, of the response series as they evolve from the presample
past (k
)Y0
) into the future.
If
YF
is a matrix,forecast
generatesnumprepaths
forecast paths, initialized by each presample response path inY0
, but the future response data, from which to condition the forecasts, is the same among all paths. Therefore,numprepaths
is the number of paths in the output argumentY
, and all paths evolve from possibly different initial conditions.If
Y0
is a matrix,forecast
initializes each response path (page) inYF
using the corresponding presample response inY0
. Therefore,numpaths
is the number of paths inYF
, and all paths in the output argumentY
derive from common initial conditions.Otherwise,
numpaths
is the minimum betweennumprepaths
and the number of pages inYF
, andforecast
appliesY0(:,:,
to initialize forecasting pathj
)
, forj
= 1,…,j
numpaths
.
Elements of YF
can be numeric scalars or missing values (indicated by
NaN
values). forecast
treats numeric scalars
as deterministic future responses that are known in advance, for example, set by policy.
forecast
forecasts responses for corresponding
NaN
values conditional on the known values.
By default, YF
is an array composed of NaN
values
indicating a complete lack of knowledge of all responses in the forecast horizon. In this
case, forecast
estimates conventional MMSE forecasts.
For more details, see Algorithms.
Example: Consider forecasting one path from a model composed of four
response series three periods into the future. Suppose that you have
prior knowledge about some of the future values of the responses, and
you want to forecast the unknown responses conditional on your
knowledge. Specify YF
as a matrix containing the
values that you know, and use NaN
for values you do
not know but want to forecast. For example, 'YF',[NaN 2 5 NaN;
NaN NaN 0.1 NaN; NaN NaN NaN NaN]
specifies that you have
no knowledge of the future values of the first and fourth response
series; you know the value for period 1 in the second response series,
but no other value; and you know the values for periods 1 and 2 in the
third response series, but not the value for period 3.
Data Types: double
PredictorVariables
— Variables to select from InSample
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 InSample
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 inInSample.Properties.VariableNames
A length
numpreds
vector of unique indices (integers) of variables to select fromInSample.Properties.VariableNames
A length
numvars
logical vector, wherePredictorVariables(
selects variablej
) = true
fromj
InSample.Properties.VariableNames
, andsum(PredictorVariables)
isnumpreds
Regardless, selected predictor variable
corresponds to the coefficients
j
Mdl.Beta(:,
.j
)
PredictorVariables
applies only when you specify
InSample
.
The selected variables must be numeric vectors and cannot contain missing values
(NaN
).
By default, forecast
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 as
the response variables.
Data Types: double
| logical
| char
| cell
| string
Note
NaN
values inY0
andX
indicate missing values.forecast
removes missing values from the data by list-wise deletion. IfY0
is a 3-D array, thenforecast
performs these steps:Horizontally concatenate pages to form a
numpreobs
-by-numpaths*numseries
matrix.Remove any row that contains at least one
NaN
from the concatenated data.
In the case of missing observations, the results obtained from multiple paths of
Y0
can differ from the results obtained from each path individually.For missing values in
X
,forecast
removes the corresponding row from each page ofYF
. After row removal fromX
andYF
, if the number of rows is less thannumperiods
,forecast
issues an error.forecast
issues an error when selected response variables fromTbl1
and selected predictor variables fromInSample
contain any missing values.
Output Arguments
Y
— MMSE forecasts of multivariate response series
numeric matrix | numeric array
MMSE forecasts of the multivariate response series, returned as a
numobs
-by-numseries
numeric matrix or a
numobs
-by-numseries
-by-numpaths
numeric array. forecast
returns Y
only when
you supply presample data Y0
as a numeric matrix or array.
Y
represents the continuation of the presample responses in
Y0
.
Each row is a time point in the simulation horizon. Specifically, row
j
contains the j
-period-ahead
forecasts. Values in a row, among all pages, occur simultaneously. The last row contains
the latest forecasted values.
Each column corresponds to the response series name in
Mdl.SeriesNames
.
Pages correspond to separate, independently forecasted paths.
If you specify future responses for conditional forecasting using the YF
name-value argument, the known values in YF
appear in the same
positions in Y
. However, Y
contains forecasted
values for the missing observations in YF
.
Tbl2
— MMSE forecasts of multivariate response series and other variables
table | timetable
Since R2022b
MMSE forecasts of multivariate response series and other variables, returned as a
table or timetable, the same data type as Tbl1
.
forecast
returns Tbl2
only when you
supply the inputs Tbl1
.
Tbl2
contains the following variables:
The forecasted response paths within the
numperiods
length forecast horizon of the selected response series yt. Each forecasted response variable inTbl2
is anumperiods
-by-numpaths
numeric matrix, wherenumpaths
depends on the number of response paths in the specified presample or future sample data (seeTbl1
orInSample
). Each row corresponds to a time in the forecast horizon and each column corresponds to a separate path.forecast
names the forecasted response variableResponseK
. For example, ifResponseK
_ResponsesMdl.Series(
isK
)GDP
,Tbl2
contains a variable for the corresponding forecasted response with the nameGDP_Responses
. If you specifyResponseVariables
,
isResponseK
ResponseVariable(
. Otherwise,K
)
isResponseK
PresampleResponseVariable(
.K
)If you specify
InSample
, all specified future response variables.
If Tbl2
is a timetable, the following conditions hold:
The row order of
Tbl2
, either ascending or descending, matches the row order ofInSample
, when you specify it. If you do not specifyInSample
, the row order ofTbl2
is the same as the row orderTbl1
.If you specify
InSample
, row timesTbl2.Time
areInSample.Time(1:numperiods)
. Otherwise,Tbl2.Time(1)
is the next time afterTbl1(end)
relative to the sampling frequency, andTbl2.Time(2:numperiods)
are the following times relative to the sampling frequency.
YMSE
— MSE matrices of forecasted responses
cell vector of numeric matrices
MSE matrices of the forecasted responses, returned as a numperiods
-by-1
cell vector of numseries
-by-numseries
numeric
matrices. Cells of YMSE
compose a time series of forecast error
covariance matrices. Cell j
contains the
j
-period-ahead MSE matrix.
YMSE
is identical for all paths.
Because forecast
treats predictor variables in X
as exogenous and nonstochastic, YMSE
reflects the error covariance
associated with the autoregressive component of the input model Mdl
only.
Algorithms
forecast
estimates unconditional forecasts using the equationwhere t = 1,...,
numperiods
.forecast
filters anumperiods
-by-numseries
matrix of zero-valued innovations throughMdl
.forecast
uses specified presample innovations (Y0
orTbl1
) wherever necessary.forecast
estimates conditional forecasts using the Kalman filter.forecast
represents the VEC modelMdl
as a state-space model (ssm
model object) without observation error.forecast
filters the forecast dataYF
through the state-space model. At period t in the forecast horizon, any unknown response iswhere s < t, is the filtered estimate of y from period s in the forecast horizon.
forecast
uses specified presample values inY0
orTbl1
for periods before the forecast horizon.
The way
forecast
determinesnumpaths
, the number of paths (pages) in the output argumentY
, or the number of paths (columns) in the forecasted response variables in the output argumentTbl2
, depends on the forecast type.If you estimate unconditional forecasts, which means you do not specify the
YF
name-value argument, orInSample
andResponseVariables
name-value arguments,numpaths
is the number of paths in theY0
orTbl1
input argument.If you estimate conditional forecasts and the presample data
Y0
and future sample dataYF
, or response variables inTbl1
andInSample
have more than one path,numpaths
is the fewest number of paths between the presample and future sample response data. Consequently,forecast
uses only the firstnumpaths
paths of each response variable for each input.If you estimate conditional forecasts and either
Y0
orYF
, or response variables inTbl1
orInSample
have one path,numpaths
is the number of pages in the array with the most pages.forecast
uses the variables with one path to produce each output path.
forecast
sets the time origin of models that include linear time trends t0 tonumpreobs
–Mdl.P
(after removing missing values), wherenumpreobs
is the number of presample observations. Therefore, the times in the trend component are t = t0 + 1, t0 + 2,..., t0 +numpreobs
. This convention is consistent with the default behavior of model estimation in whichestimate
removes the firstMdl.P
responses, reducing the effective sample size. Althoughforecast
explicitly uses the firstMdl.P
presample responses inY0
orTbl1
to initialize the model, the total number of usable observations determines t0. An observation inY0
is usable if it does not contain aNaN
.
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 R2017bR2022b: forecast
accepts input data in tables and timetables, and return results in tables and timetables
In addition to accepting input data in numeric arrays,
forecast
accepts input data in tables and timetables. forecast
chooses default series on which to operate, but you can use the following name-value arguments to select variables.
PresampleResponseVariables
specifies the response series names in the input presample response data.Insample
specifies the table or regular timetable of future response and predictor data to compute conditional forecasts.ResponseVariables
specifies the response series names inInSample
.PredictorVariables
specifies the predictor series inInSample
for a model regression component.
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