December 27, 2011 - This package computes impulse responses with Monte-Carlo confidence bands for a structural VAR.

This script illustrates how to work with Structural Vector Autoregression (SVAR) models with a simple example. Results are produced using the VAR toolbox designed by Ambrogio Cesa-Bianchi, available at https://sites.google.com/site/ambropo/MatlabCodes.

The aim of the class is to introduce the students to Vector Autoregression (VAR) and Structural Vector Autoregression (SVAR) models. The focus is on the basic concepts and practical implementation of SVARs. As such, important aspects like the estimation of these models or more advanced identification techniques are not covered. Slides.pdf: Trends and cycles, Introduction to VAR and SVAR models, Recursive (short-run) restriction, Example. SVAR_demo.mlx: MATLAB implementation of the example in the slides.

For example, you can specify values of autoregressive coefficient matrices or which lags have nonzero coefficient matrices. Whereas the varm function requires the inputs numseries and p when you use the shorthand syntax, the function must be able to infer these structural characteristics from the values you supply when you use the longhand syntax.

The syntax of a structure is "The Variable Name" followed by a "period" followed by the "field".

The code is in svarexcode.m, and it requires the matlab data file svardata.mat. It · also requires some auxiliary programs that are available on the course website. The code estimates one structural VAR that is overidentified by the order condition,

Row j contains the coefficients in the equation of response variable j, and column k contains the coefficients of exogenous variable k among all equations. δ is an n-by-1 vector of linear time-trend values. εt is an n-by-1 vector of random Gaussian innovations, each with a mean of 0 and collectively an n-by-n covariance matrix Σ. For t ≠ s, εt and εs are independent. Θk is an n-by-n matrix of MA coefficients, where k = 1,...,q and Θq is not a matrix containing only zeros. Φ0 and Θ0 are the AR and MA structural coefficients, respectively.

VAR model, specified as a varm model object returned by estimate, varm, or varm (a vecm function). ... Model summary, returned as a structure array or a varm model object.

EstMdls is an 8-by-1 vector of varm objects; EstMdls(j) represents the estimated VAR model corresponding to the model structure Mdls(j). You can access parameter estimates by using dot notation.

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EstMdl = estimate(Mdl,Y) returns the fully specified, estimated VAR(p) model EstMdl.

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We want to make a variable, called employee, that stores different types of information about an employee (name, office, code, etc.) together: employee.name='Sue'; employee.office='31B'; employee.tags={'supervisor','logistics','Chapel Hill'}; employee.code=0375; ... Matlab will return the fields and their values. The second important thing to note is that we access the fields within a structure via the period (|.|) operator.

This MATLAB function returns the variance of the elements of A along the first array dimension whose size does not equal 1.

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.

Matlab's irf() function allows for the calculation of confidence bands. In the above output, these confidence bands, stored in the objects 'lower' and 'upper', are calculated for a 95% confidence region through a bootstap procedure with 100 Monte Carlo replicas - these parameters are set by default but can be modified by the user. 2 · What portion of the variance of the VAR's forecast errors, at a given horizon h, is due to a given structural shock in $\varepsilon_t$ ?

Similar to the integers and strings we have been using so far, a structure is another data type. In this case, however, it stores other data types as well, and organizes them into containers called fields. The structure we just created, Variables, has three fields.

The key components of a varm object include the number of time series (response-variable dimensionality) and the order of the multivariate autoregressive polynomial (p) because they completely specify the model structure. Other model components include a regression component to associate the same exogenous predictor variables to each response series, and constant and time trend terms.

A structure is a MATLAB data type in which the whole object has a name, just like regular and cell arrays. However, unlike cells and regular arrays, the elements of a structure object can be accessed by their assigned field names instead of a numerical index, as in numeric arrays and cells.

This example simulates data from an arbitrary 3-D VAR(2) model, and fits a VAR(2) model to the simulated data.