# irf

Generate vector autoregression (VAR) model impulse responses

## Syntax

``Response = irf(Mdl)``
``Response = irf(Mdl,Name,Value)``
``[Response,Lower,Upper] = irf(___)``
``Tbl = irf(___)``

## Description

The `irf` function returns the dynamic response, or the impulse response function (IRF), to a one-standard-deviation shock to each variable in a VAR(p) model. A fully specified `varm` model object characterizes the VAR model.

To estimate or plot the IRF of a dynamic linear model characterized by structural, autoregression, or moving average coefficient matrices, see `armairf`.

IRFs trace the effects of an innovation shock to one variable on the response of all variables in the system. In contrast, the forecast error variance decomposition (FEVD) provides information about the relative importance of each innovation in affecting all variables in the system. To estimate the FEVD of a VAR model characterized by a `varm` model object, see `fevd`.

You can supply optional data, such as a presample, as a numeric array, table, or timetable. However, all specified input data must be the same data type. When the input model is estimated (returned by `estimate`), supply the same data type as the data used to estimate the model. The data type of the outputs matches the data type of the specified input data.

example

````Response = irf(Mdl)` returns a numeric array containing the orthogonalized IRF of the response variables that compose the VAR(p) model `Mdl`, characterized by a fully specified `varm` model object. `irf` shocks variables at time 0, and returns the IRF for times 0 through 19.If `Mdl` is an estimated model (returned by `estimate`) fit to a numeric matrix of input response data, this syntax applies.```

example

````Response = irf(Mdl,Name,Value)` `irf` returns numeric arrays when all optional input data are numeric arrays. For example, `irf(Mdl,NumObs=10,Method="generalized")` specifies estimating a generalized IRF for 10 time points starting at time 0, during which `irf` applies the shock.If `Mdl` is an estimated model fit to a numeric matrix of input response data, this syntax applies.```

example

````[Response,Lower,Upper] = irf(___)` returns numeric arrays of lower `Lower` and upper `Upper` 95% confidence bounds for confidence intervals on the true IRF, for each period and variable in the IRF, using any input argument combination in the previous syntaxes. By default, `irf` estimates confidence bounds by conducting Monte Carlo simulation.If `Mdl` is an estimated model fit to a numeric matrix of input response data, this syntax applies.If `Mdl` is a custom `varm` model object (an object not returned by `estimate` or modified after estimation), `irf` can require a sample size for the simulation `SampleSize` or presample responses `Y0`.```

example

````Tbl = irf(___)` returns the table or timetable `Tbl` containing the IRFs and, optionally, corresponding 95% confidence bounds, of the response variables that compose the VAR(p) model `Mdl`. The IRF of the corresponding response is a variable in `Tbl` containing a matrix with columns corresponding to the variables in the system shocked at time 0.If you set at least one name-value argument that controls the 95% confidence bounds on the IRF, `Tbl` also contains a variable for each of the lower and upper bounds. For example, `Tbl` contains confidence bounds when you set the `NumPaths` name-value argument.If `Mdl` is an estimated model fit to a table or timetable of input response data, this syntax applies.```

## Examples

collapse all

Fit a 4-D VAR(2) model to Danish money and income rate series data in a numeric matrix. Then, estimate and plot the orthogonalized IRF from the estimated model.

Load the Danish money and income data set.

`load Data_JDanish`

For more details on the data set, enter `Description` at the command line.

Assuming that the series are stationary, create a `varm` model object that represents a 4-D VAR(2) model. Specify the variable names.

```Mdl = varm(4,2); Mdl.SeriesNames = DataTable.Properties.VariableNames;```

`Mdl` is a `varm` model object specifying the structure of a 4-D VAR(2) model; it is a template for estimation.

Fit the VAR(2) model to the numeric matrix of time series data `Data`.

`Mdl = estimate(Mdl,Data);`

`Mdl` is a fully specified `varm` model object representing an estimated 4-D VAR(2) model.

Estimate the orthogonalized IRF from the estimated VAR(2) model.

`Response = irf(Mdl);`

`Response` is a 20-by-4-by-4 array representing the IRF of `Mdl`. Rows correspond to consecutive time points from time 0 to 19, columns correspond to variables receiving a one-standard-deviation innovation shock at time 0, and pages correspond to responses of variables to the variable being shocked. `Mdl.SeriesNames` specifies the variable order.

Display the IRF of the bond rate (variable 3, `IB`) when the log of real income (variable 2, `Y`) is shocked at time 0.

`Response(:,2,3)`
```ans = 20×1 0.0018 0.0048 0.0054 0.0051 0.0040 0.0029 0.0019 0.0011 0.0006 0.0003 ⋮ ```

Plot the IRFs of all series on separate plots by passing the estimated AR coefficient matrices and innovations covariance matrix of `Mdl` to `armairf`.

`armairf(Mdl.AR,[],InnovCov=Mdl.Covariance);`

Each plot shows the four IRFs of a variable when all other variables are shocked at time 0. `Mdl.SeriesNames` specifies the variable order.

Consider the 4-D VAR(2) model in Specify Data in Numeric Matrix When Plotting IRF. Estimate the generalized IRF of the system for 50 periods.

Load the Danish money and income data set, then estimate the VAR(2) model.

```load Data_JDanish Mdl = varm(4,2); Mdl.SeriesNames = DataTable.Properties.VariableNames; Mdl = estimate(Mdl,DataTable.Series);```

Estimate the generalized IRF from the estimated VAR(2) model.

`Response = irf(Mdl,Method="generalized",NumObs=50);`

`Response` is a 50-by-4-by-4 array representing the generalized IRF of `Mdl`.

Plot the generalized IRF of the bond rate when real income is shocked at time 0.

```figure; plot(0:49,Response(:,2,3)) title("IRF of IB When Y Is Shocked") xlabel("Observation Time") ylabel("Response") grid on```

The bond rate fades slowly when real income is shocked at time 0.

Fit a 4-D VAR(2) model to Danish money and income rate series data in a numeric matrix. Then, estimate and plot the orthogonalized IRF from the estimated model.

Load the Danish money and income data set.

`load Data_JDanish`

The data set includes four time series in the timetable `DataTimeTable`. For more details on the data set, enter `Description` at the command line.

Assuming that the series are stationary, create a `varm` model object that represents a 4-D VAR(2) model. Specify the variable names.

```Mdl = varm(4,2); Mdl.SeriesNames = DataTimeTable.Properties.VariableNames;```

`Mdl` is a `varm` model object specifying the structure of a 4-D VAR(2) model; it is a template for estimation.

Fit the VAR(2) model to the data set.

`EstMdl = estimate(Mdl,DataTimeTable);`

`Mdl` is a fully specified `varm` model object representing an estimated 4-D VAR(2) model.

Estimate the orthogonalized IRF and corresponding 95% confidence intervals from the estimated VAR(2) model. To return confidence intervals, you must set a name-value argument that controls confidence intervals, for example, `Confidence`. Set `Confidence` to `0.95`.

```rng(1); % For reproducibility Tbl = irf(EstMdl,Confidence=0.95); size(Tbl)```
```ans = 1×2 20 12 ```
`Tbl`
```Tbl=20×12 timetable Time M2_IRF Y_IRF IB_IRF ID_IRF M2_IRF_LowerBound Y_IRF_LowerBound IB_IRF_LowerBound ID_IRF_LowerBound M2_IRF_UpperBound Y_IRF_UpperBound IB_IRF_UpperBound ID_IRF_UpperBound ___________ _____________________________________________________ _______________________________________________________ ___________________________________________________ _______________________________________________________ _______________________________________________________ _______________________________________________________ ______________________________________________________ _______________________________________________________ _____________________________________________________ ___________________________________________________ ____________________________________________________ ___________________________________________________ 01-Jul-1974 0.025385 0.011979 -0.0030299 -0.00029592 0 0.017334 0.0017985 -0.00060941 0 0 0.0072384 0.0011766 0 0 0 0.0047648 0.017729 0.0062931 -0.0046236 -0.0012223 0 0.012192 -0.00017736 -0.0016259 0 0 0.0046542 -0.00050518 0 0 0 0.00322 0.02713 0.016442 -0.00074788 0.0010392 0 0.019303 0.0037597 0.00088073 0 0 0.0080468 0.002298 0 0 0 0.0049936 01-Oct-1974 0.019597 0.017604 -0.0024059 -0.00052605 0.0022668 0.014588 0.0047558 0.00037877 -0.011013 -0.0010791 0.0096387 0.0036033 -0.0014511 -0.0044686 -1.6666e-05 0.0043446 0.0088617 0.0075327 -0.0054155 -0.0024484 -0.0045762 0.0055602 0.00085487 -0.0014949 -0.015821 -0.0059937 0.0044639 0.00097209 -0.0072913 -0.0092174 -0.0021639 0.001595 0.024652 0.020779 0.00056005 0.0014407 0.0069932 0.017056 0.0070093 0.0021859 -0.0048396 0.0046483 0.010957 0.0051531 0.0043279 0.00045888 0.0017342 0.0050611 01-Jan-1975 0.023011 0.01432 -0.0014045 -0.00051972 -0.0043703 0.011341 0.005376 0.0021556 -0.019326 -0.0046725 0.010182 0.004395 0.0017534 -0.0060605 -0.0011601 0.002806 0.011414 0.0045521 -0.004882 -0.0024604 -0.01291 -0.00094466 -0.00017957 -0.00068506 -0.021662 -0.011 0.0035604 0.0011887 -0.0069887 -0.01257 -0.0045524 -0.00041328 0.029914 0.016987 0.0018906 0.0011012 0.0051311 0.013943 0.0080383 0.0041762 -0.0068581 0.003094 0.012053 0.0057324 0.0082731 0.0013124 0.0025533 0.0042406 01-Apr-1975 0.019864 0.01344 -0.0014933 -4.4254e-06 -0.0078183 0.0062247 0.0050662 0.0027626 -0.025548 -0.0070059 0.009493 0.0043074 0.0045597 -0.004589 -0.002223 0.0011591 0.0063539 0.0025976 -0.0054684 -0.0021938 -0.017355 -0.0064979 -0.00123 0.00010743 -0.029292 -0.013974 0.0015562 0.00077995 -0.0081444 -0.011287 -0.0062556 -0.0022002 0.028924 0.015926 0.0014925 0.0012851 0.0059606 0.011209 0.0080886 0.0047651 -0.0063978 0.0028959 0.012097 0.0057966 0.014418 0.0038715 0.0033635 0.0023838 01-Jul-1975 0.019419 0.011244 -0.0015969 -3.4305e-05 -0.010187 0.0028147 0.0039998 0.0026985 -0.029124 -0.0084906 0.0084798 0.0037481 0.0077423 -0.0025063 -0.0026634 -0.00013404 0.005327 -0.00049621 -0.0054593 -0.0021455 -0.022716 -0.0081187 -0.0028733 -0.00042378 -0.034992 -0.015799 0.00056474 0.00020164 -0.0078458 -0.010846 -0.0068023 -0.0035519 0.029407 0.014405 0.001692 0.0015767 0.0087664 0.010134 0.0075162 0.0043343 -0.0057691 0.0018746 0.012047 0.0055231 0.020043 0.0064893 0.0029097 0.001343 01-Oct-1975 0.018532 0.01016 -0.0019436 -0.00013964 -0.010448 0.00049217 0.0028587 0.0021692 -0.031107 -0.0092773 0.0074784 0.0031474 0.0097242 -0.00029442 -0.0026358 -0.00082626 0.001818 -0.0004023 -0.0058779 -0.0022755 -0.023734 -0.0077482 -0.0042795 -0.00098779 -0.038594 -0.016547 -0.00011469 -0.00064435 -0.0095859 -0.0070065 -0.0059945 -0.0034737 0.030136 0.013705 0.0016763 0.0017479 0.011566 0.0082498 0.0057995 0.0033017 -0.0044473 0.0013279 0.011642 0.0050999 0.022978 0.0090112 0.0027824 0.0014644 01-Jan-1976 0.018426 0.0092868 -0.0022099 -0.00036763 -0.0097658 -0.00071481 0.0018519 0.0015618 -0.032019 -0.0098054 0.0066319 0.002661 0.010735 0.0013489 -0.0022958 -0.001066 0.00094487 -0.0010919 -0.0057706 -0.0026426 -0.023367 -0.0091729 -0.0049467 -0.0014946 -0.042002 -0.016681 -0.00084973 -0.0010919 -0.010268 -0.0059067 -0.0046107 -0.0030697 0.030831 0.014467 0.0020979 0.0014645 0.013995 0.0070356 0.0042604 0.0028084 -0.0042527 0.0021613 0.010919 0.0047273 0.024747 0.010363 0.0021819 0.00073538 01-Apr-1976 0.018347 0.0088274 -0.002408 -0.00055954 -0.0085284 -0.0013241 0.0011035 0.001029 -0.032301 -0.010245 0.0059331 0.0023208 0.010834 0.0024299 -0.0018575 -0.0010181 0.00088792 -0.00087384 -0.0057043 -0.0027239 -0.022321 -0.0095626 -0.0046415 -0.0016955 -0.044772 -0.015403 -0.0014757 -0.00074988 -0.010505 -0.0057904 -0.0042961 -0.0025666 0.030874 0.015059 0.0021942 0.0012385 0.01511 0.0075202 0.0041588 0.0022807 -0.0018926 0.0020667 0.0099711 0.0041106 0.027227 0.010447 0.0026554 0.0011517 01-Jul-1976 0.018349 0.0085007 -0.0025003 -0.00070381 -0.0072023 -0.0015783 0.00059083 0.00064004 -0.032139 -0.010662 0.0053454 0.0020863 0.010389 0.0030152 -0.0014396 -0.00084984 0.00081535 -0.00056917 -0.0054989 -0.0027904 -0.020095 -0.0090132 -0.003905 -0.0019258 -0.046259 -0.016859 -0.0018945 -0.0010315 -0.0088981 -0.0047543 -0.0042238 -0.0017878 0.030856 0.014085 0.0023174 0.0013691 0.017564 0.0079639 0.0040872 0.0017655 -0.001588 0.0020803 0.0088559 0.0036667 0.02646 0.010048 0.0026267 0.0012534 01-Oct-1976 0.018261 0.0082681 -0.0025175 -0.00078515 -0.0059528 -0.0016715 0.000265 0.00038128 -0.031642 -0.01103 0.0048331 0.0019104 0.0096565 0.0032746 -0.0011032 -0.00066034 -0.0011251 -0.00075733 -0.0050727 -0.0026529 -0.019957 -0.0091831 -0.0037003 -0.0015265 -0.046571 -0.018406 -0.0023099 -0.0010146 -0.0087836 -0.0043276 -0.0035229 -0.0015206 0.031925 0.013921 0.0023499 0.0013534 0.017285 0.0080779 0.0037871 0.0016071 0.00034958 0.0018116 0.0082019 0.0031525 0.025188 0.0098717 0.0023763 0.001372 01-Jan-1977 0.018088 0.0080578 -0.0024803 -0.00082067 -0.004866 -0.0016783 6.7351e-05 0.00021975 -0.030874 -0.011308 0.0043776 0.0017587 0.0088385 0.0033353 -0.00085828 -0.00049952 -0.0023654 -0.00054861 -0.0046959 -0.0023017 -0.018881 -0.0083688 -0.0036323 -0.0013151 -0.045822 -0.019143 -0.0022457 -0.00076165 -0.0096651 -0.0046474 -0.0033535 -0.001605 0.031832 0.013992 0.0023109 0.0011487 0.01618 0.0079136 0.0033058 0.0015065 0.0014169 0.0019553 0.0074233 0.0026903 0.023693 0.0092439 0.0022507 0.0010427 01-Apr-1977 0.01782 0.0078585 -0.0024126 -0.00082488 -0.0039472 -0.0016348 -4.6051e-05 0.00012029 -0.029898 -0.011467 0.0039728 0.0016154 0.0080434 0.0032901 -0.00069265 -0.00038034 -0.0041445 -0.0009108 -0.0046233 -0.0021475 -0.017842 -0.0074943 -0.0036166 -0.0012714 -0.044148 -0.019166 -0.0023327 -0.00078836 -0.010263 -0.0046773 -0.0028019 -0.001797 0.030983 0.013903 0.0022172 0.001152 0.014886 0.007478 0.0027978 0.0013601 0.0017628 0.0019984 0.0059992 0.0024748 0.023083 0.0093155 0.0021594 0.00090656 01-Jul-1977 0.01748 0.0076612 -0.0023288 -0.00081178 -0.0031853 -0.0015527 -0.00010551 5.8177e-05 -0.028778 -0.011497 0.0036184 0.001477 0.007324 0.0031905 -0.00058632 -0.00029882 -0.0055294 -0.00154 -0.0045625 -0.0019225 -0.017729 -0.00744 -0.0034336 -0.0012351 -0.041703 -0.018635 -0.0029266 -0.00097794 -0.010567 -0.004571 -0.0030526 -0.00144 0.029897 0.013651 0.002083 0.0011265 0.015725 0.0067952 0.0028631 0.0012628 0.0032766 0.0021382 0.0054534 0.00231 0.021 0.0091161 0.00197 0.000865 01-Oct-1977 0.017084 0.0074669 -0.0022386 -0.00078982 -0.0025611 -0.0014412 -0.00012965 1.837e-05 -0.027577 -0.01141 0.003315 0.0013467 0.0066953 0.0030638 -0.00052096 -0.00024515 -0.0059883 -0.0019006 -0.0045009 -0.0017518 -0.016735 -0.0070905 -0.0030476 -0.0012138 -0.03866 -0.017655 -0.0023904 -0.0010579 -0.010714 -0.0044868 -0.0030057 -0.0010935 0.028693 0.01332 0.0015194 0.0010569 0.016655 0.0061735 0.0029008 0.0010539 0.0049304 0.0023071 0.0052074 0.0022951 0.019005 0.008691 0.0018979 0.00088102 01-Jan-1978 0.016649 0.007275 -0.0021468 -0.00076393 -0.0020585 -0.0013098 -0.00013023 -6.9227e-06 -0.02635 -0.011223 0.0030608 0.0012284 0.006156 0.0029232 -0.00048251 -0.00021015 -0.0067654 -0.0016931 -0.0044291 -0.0016366 -0.016779 -0.0072429 -0.0028373 -0.0011991 -0.035195 -0.016242 -0.0025182 -0.0010499 -0.010659 -0.0043064 -0.002775 -0.0010409 0.028206 0.01294 0.0015156 0.00095142 0.016305 0.0058332 0.0026818 0.00094436 0.0054449 0.0022641 0.0053236 0.0021337 0.018369 0.0079906 0.0017187 0.00082592 01-Apr-1978 0.016187 0.0070851 -0.0020565 -0.00073641 -0.0016627 -0.0011692 -0.00011514 -2.1687e-05 -0.025141 -0.010963 0.0028515 0.001125 0.0056974 0.0027764 -0.00046133 -0.00018714 -0.0053812 -0.0015885 -0.0043428 -0.0016435 -0.016571 -0.007324 -0.0025033 -0.0011496 -0.033556 -0.015142 -0.0025535 -0.0010552 -0.010407 -0.0040787 -0.0025947 -0.0011176 0.027675 0.012605 0.0015485 0.00082164 0.015764 0.0056146 0.0024931 0.0010281 0.0054922 0.0019958 0.0054463 0.0021499 0.019244 0.0078422 0.00157 0.00071951 ⋮ ```

`Tbl` is a timetable with 20 rows, representing the periods in the IRF, and 12 variables. Each variable is a 20-by-4 matrix of the IRF or confidence bound associated with a variable in the model `EstMdl`. For example, `Tbl.M2_IRF(:,2)` is the IRF of `M2` resulting from a 1-standard-deviation shock on `01-Jul-1974` (period 0) to `Mdl.SeriesNames(2)`, which is the variable `Y`. [`Tbl.M2_IRF_LowerBound(:,2)`,`Tbl.M2_IRF_UpperBound(:,2)`] are the corresponding 95% confidence intervals.

Plot the IRF of M2 and its 95% confidence interval resulting from a 1-standard-deviation shock on `01-Jul-1974` (period 0) to `Mdl.SeriesNames(2)`, which is the variable `Y`.

```idxM2 = startsWith(Tbl.Properties.VariableNames,"M2"); M2IRF = Tbl(:,idxM2); shockIdx = 2; figure hold on plot(M2IRF.Time,M2IRF.M2_IRF(:,shockIdx),"-o") plot(M2IRF.Time,[M2IRF.M2_IRF_LowerBound(:,shockIdx) ... M2IRF.M2_IRF_UpperBound(:,shockIdx)],"-o",Color="r") legend("IRF","95% confidence interval") title('M2 IRF, Shock to Y') hold off```

Consider the 4-D VAR(2) model in Specify Data in Numeric Matrix When Plotting IRF. Estimate and plot its orthogonalized IRF and 95% Monte Carlo confidence intervals on the true IRF.

Load the Danish money and income data set, then estimate the VAR(2) model.

```load Data_JDanish Mdl = varm(4,2); Mdl.SeriesNames = DataTable.Properties.VariableNames; Mdl = estimate(Mdl,DataTable.Series);```

Estimate the IRF and corresponding 95% Monte Carlo confidence intervals from the estimated VAR(2) model.

```rng(1); % For reproducibility [Response,Lower,Upper] = irf(Mdl);```

`Response`, `Lower`, and `Upper` are 20-by-4-by-4 arrays representing the orthogonalized IRF of `Mdl` and corresponding lower and upper bounds of the confidence intervals. For all arrays, rows correspond to consecutive time points from time 0 to 19, columns correspond to variables receiving a one-standard-deviation innovation shock at time 0, and pages correspond to responses of variables to the variable being shocked. `Mdl.SeriesNames` specifies the variable order.

Plot the orthogonalized IRF with its confidence bounds of the bond rate when real income is shocked at time 0.

```irfshock2resp3 = Response(:,2,3); IRFCIShock2Resp3 = [Lower(:,2,3) Upper(:,2,3)]; figure; h1 = plot(0:19,irfshock2resp3); hold on h2 = plot(0:19,IRFCIShock2Resp3,"r--"); legend([h1 h2(1)],["IRF" "95% Confidence Interval"]) xlabel("Time Index"); ylabel("Response"); title("IRF of IB When Y Is Shocked"); grid on hold off```

The effect of the impulse to real income on the bond rate fades after 10 periods.

Consider the 4-D VAR(2) model in Specify Data in Numeric Matrix When Plotting IRF. Estimate and plot its orthogonalized IRF and 90% bootstrap confidence intervals on the true IRF.

Load the Danish money and income data set, then estimate the VAR(2) model. Return the residuals from model estimation.

```load Data_JDanish Mdl = varm(4,2); Mdl.SeriesNames = DataTable.Properties.VariableNames; [Mdl,~,~,Res] = estimate(Mdl,DataTable.Series); T = size(DataTable,1) % Total sample size```
```T = 55 ```
`n = size(Res,1) % Effective sample size`
```n = 53 ```

`Res` is a 53-by-4 array of residuals. Columns correspond to the variables in `Mdl.SeriesNames`. The `estimate` function requires `Mdl.P` = 2 observations to initialize a VAR(2) model for estimation. Because presample data (`Y0`) is unspecified, `estimate` takes the first two observations in the specified response data to initialize the model. Therefore, the resulting effective sample size is `T``Mdl.P` = 53, and rows of `Res` correspond to the observation indices 3 through `T`.

Estimate the orthogonalized IRF and corresponding 90% bootstrap confidence intervals from the estimated VAR(2) model. Draw 500 paths of length `n` from the series of residuals.

```rng(1); % For reproducibility [Response,Lower,Upper] = irf(Mdl,E=Res,NumPaths=500,Confidence=0.9);```

Plot the orthogonalized IRF with its confidence bounds of the bond rate when real income is shocked at time 0.

```irfshock2resp3 = Response(:,2,3); IRFCIShock2Resp3 = [Lower(:,2,3) Upper(:,2,3)]; figure; h1 = plot(0:19,irfshock2resp3); hold on h2 = plot(0:19,IRFCIShock2Resp3,"r--"); legend([h1 h2(1)],["IRF" "90% Confidence Interval"]) xlabel("Time Index"); ylabel("Response"); title("IRF of IB When Y Is Shocked"); grid on hold off```

The effect of the impulse to real income on the bond rate fades after 10 periods.

## Input Arguments

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VAR model, specified as a `varm` model object created by `varm` or `estimate`. `Mdl` must be fully specified.

If `Mdl` is an estimated model (returned by `estimate`) , you must supply any optional data using the same data type as the input response data, to which the model is fit.

If `Mdl` is a custom `varm` model object (an object not returned by `estimate` or modified after estimation), `irf` can require a sample size for the simulation `SampleSize` or presample responses `Y0`.

### 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: `irf(Mdl,NumObs=10,Method="generalized")` specifies estimating a generalized IRF for 10 time points starting at time 0, during which `irf` applies the shock.

Options for All IRFs

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Number of periods for which `irf` computes the IRF, specified as a positive integer. `NumObs` specifies the number of observations to include in the IRF (the number of rows in `Response` or `Tbl`).

Example: `NumObs=10` specifies the inclusion of 10 time points in the IRF starting at time 0, during which `irf` applies the shock, and ending at period 9.

Data Types: `double`

IRF computation method, specified as a value in this table.

ValueDescription
`"orthogonalized"`Compute impulse responses using orthogonalized, one-standard-deviation innovation shocks. `irf` uses the Cholesky factorization of `Mdl.Covariance` for orthogonalization.
`"generalized"`Compute impulse responses using one-standard-deviation innovation shocks.

Example: `Method="generalized"`

Data Types: `string` | `char`

Options for Confidence Bound Estimation

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Number of sample paths (trials) to generate, specified as a positive integer.

Example: `NumPaths=1000` generates `1000` sample paths from which the software derives the confidence bounds.

Data Types: `double`

Number of observations for the Monte Carlo simulation or bootstrap per sample path, specified as a positive integer.

• If `Mdl` is an estimated `varm` model object (an object returned by `estimate` and unmodified thereafter), the default is the sample size of the data to which the model is fit (see `summarize`).

• Otherwise:

• If `irf` estimates confidence bounds by conducting a Monte Carlo simulation, you must specify `SampleSize`.

• If `irf` estimates confidence bounds by bootstrapping residuals, the default is the length of the specified series of residuals (`size(Res,1)`, where `Res` is the number of residuals in `E` or `InSample`).

Example: If you specify `SampleSize=100` and do not specify the `E` name-value argument, the software estimates confidence bounds from `NumPaths` random paths of length `100` from `Mdl`.

Example: If you specify `SampleSize=100,E=Res`, the software resamples, with replacement, `100` observations (rows) from `Res` to form a sample path of innovations to filter through `Mdl`. The software forms `NumPaths` random sample paths from which it derives confidence bounds.

Data Types: `double`

Presample response data that provides initial values for model estimation during the simulation, specified as a `numpreobs`-by-`numseries` numeric matrix. Use `Y0` only in the following situations:

`numpreobs` is the number of presample observations. `numseries` is `Mdl.NumSeries`, the dimensionality of the input model.

Each row is a presample observation, and measurements in each row occur simultaneously. The last row contains the latest presample observation. `numpreobs` is the number of specified presample responses and it must be at least `Mdl.P`. If you supply more rows than necessary, `irf` uses the latest `Mdl.P` observations only.

`numseries` is the dimensionality of the input VAR model `Mdl.NumSeries`. Columns must correspond to the response variables in `Mdl.SeriesNames`.

The following situations determine the default or whether presample response data is required.

• If `Mdl` is an unmodified estimated model, `irf` sets `Y0` to the presample response data used for estimation by default (see the `Y0` name-value argument of `estimate`).

• If `Mdl` is a custom model and you return confidence bounds `Lower` or `Upper`, you must specify `Y0`.

Data Types: `double`

Presample data that provide initial values for the model `Mdl`, specified as a table or timetable with `numprevars` variables and `numpreobs` rows. Use `Presample` only in the following situations:

• You supply other optional data inputs as tables or timetables.

• `Mdl` is an estimated `varm` model object (an object returned by `estimate` and unmodified thereafter) fit to response data in a table or timetable.

Each row is a presample observation, and measurements in each row occur simultaneously. `numpreobs` must be at least `Mdl.P`. If you supply more rows than necessary, `irf` uses the latest `Mdl.P` observations only.

Each variable is a `numpreobs` numeric vector representing one path. To control presample variable selection, see the optional `PresampleResponseVariables` name-value argument.

If `Presample` is a timetable, all the following conditions must be true:

• `Presample` must represent a sample with a regular datetime time step (see `isregular`).

• The datetime vector of sample timestamps `Presample.Time` must be ascending or descending.

If `Presample` is a table, the following conditions hold:

• The last row contains the latest presample observation.

• `Presample.Properties.RowsNames` must be empty.

The following situations determine the default or whether presample response data is required.

• If `Mdl` is an unmodified estimated model, `irf` sets `Presample` to the presample response data used for estimation by default (see the `Presample` name-value argument of `estimate`).

• If `Mdl` is a custom model (for example, you modify a model after estimation by using dot notation) and you return confidence bounds in the table or timetable `Tbl`, you must specify `Presample`.

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 in `Presample.Properties.VariableNames`

• A length `numseries` vector of unique indices (integers) of variables to select from `Presample.Properties.VariableNames`

• A length `numprevars` logical vector, where `PresampleResponseVariables(j) = true ` selects variable `j` from `Presample.Properties.VariableNames`, and `sum(PresampleResponseVariables)` is `numseries`

`PresampleResponseVariables` applies only when you specify `Presample`.

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`; `irf` uses the data in selected variable `PresampleResponseVariables(j)` as a presample for `Mdl.SeriesNames(j)`.

If the number of variables in `Presample` matches `Mdl.NumSeries`, the default specifies all variables in `Presample`. If the number of variables in `Presample` exceeds `Mdl.NumSeries`, the default matches variables in `Presample` 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`

Predictor data xt for estimating the model regression component during the simulation, specified as a numeric matrix containing `numpreds` columns. Use `X` only in the following situations:

`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 `SampleSize` rows. If you supply more rows than necessary, `irf` uses only the latest observations. `irf` does not use the regression component in the presample period.

Columns correspond to individual predictor variables. All predictor variables are present in the regression component of each response equation.

To maintain model consistency when `irf` estimates the confidence bounds, a good practice is to specify predictor data when `Mdl` has a regression component. If `Mdl` is an estimated model, specify the predictor data used during model estimation (see the `X` name-value argument of `estimate`).

By default, `irf` excludes the regression component from confidence bound estimation, regardless of its presence in `Mdl`.

Data Types: `double`

Series of residuals from which to draw bootstrap samples, specified as a `numperiods`-by-`numseries` numeric matrix. `irf` assumes that `E` is free of serial correlation. Use `E` only in the following situations:

Each column is the residual series corresponding to the response series names in `Mdl.SeriesNames`.

Each row corresponds to a period in the FEVD and the corresponding confidence bounds.

If `Mdl` is an estimated `varm` model object (an object returned by `estimate`), you can specify `E` as the inferred residuals from estimation (see the `E` output argument of `estimate` or `infer`).

By default, `irf` derives confidence bounds by conducting a Monte Carlo simulation.

Data Types: `double`

Time series data containing `numvars` variables, including `numseries` variables of residuals et to bootstrap or `numpreds` predictor variables xt for the model regression component, specified as a table or timetable. Use `InSample` only in the following situations:

Each variable is a single path of observations, which `irf` applies to all `NumPaths` sample paths. If you specify `Presample` you must specify which variables are residuals and predictors, see the `ResidualVariables` and `PredictorVariables` name-value arguments.

Each row is an observation, and measurements in each row occur simultaneously. `InSample` must have at least `SampleSize` rows. If you supply more rows than necessary, `irf` uses only the latest observations.

If `InSample` is a timetable, the following conditions apply:

• `InSample` must represent a sample with a regular datetime time step (see `isregular`).

• The datetime vector `InSample.Time` must be strictly ascending or descending.

• `Presample` must immediately precede `InSample`, with respect to the sampling frequency.

If `InSample` is a table, the following conditions hold:

• The last row contains the latest observation.

• `InSample.Properties.RowsNames` must be empty.

By default, `irf` derives confidence bounds by conducting a Monte Carlo simulation and does not use model the regression component, regardless of its presence in `Mdl`.

Variables to select from `InSample` to treat as residuals for bootstrapping, specified as one of the following data types:

• String vector or cell vector of character vectors containing `numseries` variable names in `InSample.Properties.VariableNames`

• A length `numseries` vector of unique indices (integers) of variables to select from `InSample.Properties.VariableNames`

• A length `numvars` logical vector, where `ResidualVariables(j) = true` selects variable `j` from `InSample.Properties.VariableNames`, and `sum(ResidualVariables)` is `numseries`

Regardless, selected residual variable `j` is the residual series for `Mdl.SeriesNames(j)`.

The selected variables must be numeric vectors and cannot contain missing values (`NaN`).

By default, `irf` derives confidence bounds by conducting a Monte Carlo simulation.

Example: `ResidualVariables=["GDP_Residuals" "CPI_Residuals"]`

Example: `ResidualVariables=[true false true false]` or `ResidualVariable=[1 3]` selects the first and third table variables as the disturbance variables.

Data Types: `double` | `logical` | `char` | `cell` | `string`

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 in `InSample.Properties.VariableNames`

• A length `numpreds` vector of unique indices (integers) of variables to select from `InSample.Properties.VariableNames`

• A length `numvars` logical vector, where `PredictorVariables(j) = true ` selects variable `j` from `InSample.Properties.VariableNames`, and `sum(PredictorVariables)` is `numpreds`

Regardless, selected predictor variable `j` corresponds to the coefficients `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, `irf` 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`

Confidence level for the confidence bounds, specified as a numeric scalar in the interval [0,1].

For each period, randomly drawn confidence intervals cover the true response `100*Confidence`% of the time.

The default value is `0.95`, which implies that the confidence bounds represent 95% confidence intervals.

Example: `Confidence=0.9` specifies 90% confidence intervals.

Data Types: `double`

Note

• `NaN` values in `Y0`, `X`, and `E` indicate missing data. `irf` removes missing data from these arguments by list-wise deletion. Each argument, if a row contains at least one `NaN`, then `irf` removes the entire row.

List-wise deletion reduces the sample size, can create irregular time series, and can cause `E` and `X` to be unsynchronized.

• `irf` issues an error when any table or timetable input contains missing values.

## Output Arguments

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IRF of each variable, returned as a `numobs`-by-`numseries`-by-`numseries` numeric array. `numobs` is the value of `NumObs`. Columns and pages correspond to the response variables in `Mdl.SeriesNames`.

`irf` returns `Response` only in the following situations:

• You supply optional data inputs as numeric matrices.

• `Mdl` is an estimated model fit to a numeric matrix of response data.

```Response(t + 1,j,k)``` is the impulse response of variable `k` to a one-standard-deviation innovation shock to variable `j` at time 0, for `t` = 0, 1, ..., `numObs` – 1, `j` = 1,2,...,`numseries`, and `k` = 1,2,...,`numseries`.

Lower confidence bounds, returned as a `numobs`-by-`numseries`-by-`numseries` numeric array. Elements of `Lower` correspond to elements of `Response`.

`irf` returns `Lower` only in the following situations:

• You supply optional data inputs as numeric matrices.

• `Mdl` is an estimated model fit to a numeric matrix of response data.

```Lower(t + 1,j,k)``` is the lower bound of the `100*Confidence`-th percentile interval on the true impulse response of variable `k` to a one-standard-deviation innovation shock to variable `j` at time 0.

Upper confidence bounds, returned as a `numobs`-by-`numseries`-by-`numseries` numeric array. Elements of `Upper` correspond to elements of `Response`.

`irf` returns `Upper` only in the following situations:

• You supply optional data inputs as numeric matrices.

• `Mdl` is an estimated model fit to a numeric matrix of response data.

```Upper(t + 1,j,k)``` is the upper bound of the `100*Confidence`-th percentile interval on the true impulse response of variable `k` to a one-standard-deviation innovation shock to variable `j` at time 0.

IRF and confidence bounds, returned as a table or timetable with `numobs` rows. `irf` returns `Tbl` only in the following situations:

• You supply optional data inputs as tables or timetables.

• `Mdl` is an estimated model fit to response data in a table or timetable.

Regardless, the data type of `Tbl` is the same as the data type of specified data.

`Tbl` contains the following variables:

• The IRF of each series in yt. Each IRF variable in `Tbl` is a `numobs`-by-`numseries` numeric matrix, where `numobs` is the value of `NumObs` and `numseries` is the value of `Mdl.NumSeries`. `irf` names the IRF of response variable `ResponseJ` in `Mdl.SeriesNames` `ResponseJ_IRF`. For example, if `Mdl.Series(j)` is `GDP`, `Tbl` contains a variable for the corresponding IRF with the name `GDP_IRF`.

`ResponseJ_IRF``(t + 1,k)` is the impulse response of variable `ResponseJ` to a one-standard-deviation innovation shock to variable `k` at time 0, for `t` = 0, 1, ..., `numObs` – 1, `J` = 1,2,...,`numseries`, and `k` = 1,2,...,`numseries`.

• The lower and upper confidence bounds on the true IRF of the response series, when you set at least one name-value argument that controls the confidence bounds. Each confidence bound variable in `Tbl` is a `numobs`-by-`numseries` numeric matrix. `ResponseJ_IRF_LowerBound` and `ResponseJ_IRF_UpperBound` are the names of the lower and upper bound variables, respectively, of the confidence interval on the IRF of response variable `Mdl.SeriesNames(J)` = `ResponseJ` . For example, if `Mdl.SeriesNames(j)` is `GDP`, `Tbl` contains variables for the corresponding lower and upper bounds of the confidence interval with the name `GDP_IRF_LowerBound` and `GDP_IRF_UpperBound`.

(`ResponseJ_IRF_LowerBound(t,k)`, `ResponseJ_IRF_UpperBound(t,k)`) is the 95% confidence interval on the IRF of response variable `ResponseJ` attributable to a one-standard-deviation innovation shock to variable `k` at time `t`, for `t` = 0,1,…,`numobs` – 1, `J` = 1,2,...,`numseries`, and `k` = 1,2,...,`numseries`.

If `Tbl` is a timetable, the row order of `Tbl`, either ascending or descending, matches the row order of `InSample`, when you specify it. If you do not specify `InSample` and you specify `Presample`, the row order of `Tbl` is the same as the row order `Presample`.

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### Impulse Response Function

An impulse response function (IRF) of a time series model (or dynamic response of the system) measures the changes in the future responses of all variables in the system when a variable is shocked by an impulse. In other words, the IRF at time t is the derivative of the responses at time t with respect to an innovation at time t0 (the time that innovation was shocked), tt0.

Consider a `numseries`-D VAR(p) model for the multivariate response variable yt. In lag operator notation, the infinite lag MA representation of yt is:

`$\begin{array}{c}{y}_{t}={\Phi }^{-1}\left(L\right)\left(c+\beta {x}_{t}+\delta t\right)+{\Phi }^{-1}\left(L\right){\epsilon }_{t}\\ =\Omega \left(L\right)\left(c+\beta {x}_{t}+\delta t\right)+\Omega \left(L\right){\epsilon }_{t}.\end{array}$`

The general form of the IRF of yt shocked by an impulse to variable j by one standard deviation of its innovation m periods into the future is:

`${\psi }_{j}\left(m\right)={C}_{m}{e}_{j}.$`

• ej is a selection vector of length `numseries` containing a 1 in element j and zeros elsewhere.

• For the orthogonalized IRF, ${C}_{m}={\Omega }_{m}P,$ where P is the lower triangular factor in the Cholesky factorization of Σ, and Ωm is the lag m coefficient of Ω(L).

• For the generalized IRF, ${C}_{m}={\sigma }_{j}^{-1}{\Omega }_{m}\Sigma ,$ where σj is the standard deviation of innovation j.

• The IRF is free of the model constant, regression component, and time trend.

### Vector Autoregression Model

A vector autoregression (VAR) model is a stationary multivariate time series model consisting of a system of m equations of m distinct response variables as linear functions of lagged responses and other terms.

A VAR(p) model in difference-equation notation and in reduced form is

`${y}_{t}=c+{\Phi }_{1}{y}_{t-1}+{\Phi }_{2}{y}_{t-2}+...+{\Phi }_{p}{y}_{t-p}+\beta {x}_{t}+\delta t+{\epsilon }_{t}.$`

• yt is a `numseries`-by-1 vector of values corresponding to `numseries` response variables at time t, where t = 1,...,T. The structural coefficient is the identity matrix.

• c is a `numseries`-by-1 vector of constants.

• Φj is a `numseries`-by-`numseries` matrix of autoregressive coefficients, where j = 1,...,p and Φp is not a matrix containing only zeros.

• xt is a `numpreds`-by-1 vector of values corresponding to `numpreds` exogenous predictor variables.

• β is a `numseries`-by-`numpreds` matrix of regression coefficients.

• δ is a `numseries`-by-1 vector of linear time-trend values.

• εt is a `numseries`-by-1 vector of random Gaussian innovations, each with a mean of 0 and collectively a `numseries`-by-`numseries` covariance matrix Σ. For ts, εt and εs are independent.

Condensed and in lag operator notation, the system is

`$\Phi \left(L\right){y}_{t}=c+\beta {x}_{t}+\delta t+{\epsilon }_{t},$`

where $\Phi \left(L\right)=I-{\Phi }_{1}L-{\Phi }_{2}{L}^{2}-...-{\Phi }_{p}{L}^{p}$, Φ(L)yt is the multivariate autoregressive polynomial, and I is the `numseries`-by-`numseries` identity matrix.

## Algorithms

• If `Method` is `"orthogonalized"`, then the resulting IRF depends on the order of the variables in the time series model. If `Method` is `"generalized"`, then the resulting IRF is invariant to the order of the variables. Therefore, the two methods generally produce different results.

• If `Mdl.Covariance` is a diagonal matrix, then the resulting generalized and orthogonalized IRFs are identical. Otherwise, the resulting generalized and orthogonalized IRFs are identical only when the first variable shocks all variables (for example, all else being the same, both methods yield the same value of `Response(:,1,:)`).

• The predictor data in `X` or `InSample` represents a single path of exogenous multivariate time series. If you specify `X` or `InSample` and the model `Mdl` has a regression component (`Mdl.Beta` is not an empty array), `irf` applies the same exogenous data to all paths used for confidence interval estimation.

• `irf` conducts a simulation to estimate the confidence bounds `Lower` and `Upper` or associated variables in `Tbl`.

• If you do not specify residuals by supplying `E` or using `InSample`, `irf` conducts a Monte Carlo simulation by following this procedure:

1. Simulate `NumPaths` response paths of length `SampleSize` from `Mdl`.

2. Fit `NumPaths` models that have the same structure as `Mdl` to the simulated response paths. If `Mdl` contains a regression component and you specify predictor data by supplying `X` or using `InSample`, then `irf` fits the `NumPaths` models to the simulated response paths and the same predictor data (the same predictor data applies to all paths).

3. Estimate `NumPaths` IRFs from the `NumPaths` estimated models.

4. For each time point t = 0,…,`NumObs`, estimate the confidence intervals by computing 1 – `Confidence` and `Confidence` quantiles (upper and lower bounds, respectively).

• Otherwise, `irf` conducts a nonparametric bootstrap by following this procedure:

1. Resample, with replacement, `SampleSize` residuals from `E` or `InSample`. Perform this step `NumPaths` times to obtain `NumPaths` paths.

2. Center each path of bootstrapped residuals.

3. Filter each path of centered, bootstrapped residuals through `Mdl` to obtain `NumPaths` bootstrapped response paths of length `SampleSize`.

4. Complete steps 2 through 4 of the Monte Carlo simulation, but replace the simulated response paths with the bootstrapped response paths.

## References

[1] Hamilton, James D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.

[2] Lütkepohl, Helmut. New Introduction to Multiple Time Series Analysis. New York, NY: Springer-Verlag, 2007.

[3] Pesaran, H. H., and Y. Shin. "Generalized Impulse Response Analysis in Linear Multivariate Models." Economic Letters. Vol. 58, 1998, pp. 17–29.

## Version History

Introduced in R2019a