In a previous post the dynamics of U.S. macroeconomic variables were estimated using a Vector Autoregression. In that standard VAR estimation every equation can be estimated as a stand alone regression, but there some specification issues and violations of the Classical Linear Regression Model are present. In this post a Structural Vector Autoregression will be identified and estimated using STATA. Special restrictions will be based on the contemporaneous affects of macroeconomic variables to get better estimates Germany’s long range dynamics. The restrictions on contemporaneous interactions among variables will be lower triangular and will yield what is called a Cholesky decomposition of the SVAR. At the end of this post a analysis will be calculated that will explain the short term impact of changes in income and investment on consumption in the short-term.

**Data and Variables**

The data used belong from the STATA data library and is based on work done by Lutkephol(1993) and contains quarterly data from Germany from the time period of 1960q1 to 1982q2. There are 3 macroeconomic variables that will be analyzed are investment, income, and consumption. All 3 variables will be at the first difference of the logs level to model elasticities and ensure a stationary SVAR and to ease in the interpretation of the Cholesky Decomposition.

Data Source: http://www.stata-press.com/data/r11/lutkepho12

**Restrictions on Contemporaneous Matrix Following A Cholesky Decomposition
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The Cholesky restrictions will be placed on shi system by first defining the contemporenous matrix in STATA. Creating these matrices in STATA is fairly simple; numbers in the matrices are restrictions and “.” ‘s are parameters that the program is free to estimate:

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1Since we have that y = (investment, income, consumption), the A matrix above imposes these restrictions on the contemporaneous interactions among these variables:

**Matrix A r1:**Percentage changes in investments are not contemporaneously affected by consumption or income**Matrix A r2:**Percentage changes in income is affected by contemporaneous changes in investments but not consumption.**Matrix A r3:**Finally we assume that percentages changes in consumption are affected by contemporaneous changes in both investments and income.

**Matrix B**: Is defined as a matrix which is restricted to be diagonal; this matrix represents the weights given to the error terms in the structural VAR

One can argue that the restrictions imposed by this decomposition are not optimum or that there might be a better way to select them. Theoretical foundations in both macroeconomics and microeconomics have been used to identify this A matrix. Given the fluidity of economic ideas and theories a better way to restrict the A matrix is possible.

**Estimating SVAR**

The Iteration Log gives a step by step account of the iterations and the log likelihood estimates that corresponds to these units. The next section displays the constraints imposed on the on the A and B matrices. Finally the estimates for the A and B matrices are displayed just below the a header which contains the sample size, no. of obs, Log likelihood and and message indicating that the model is “Exactly identified”.

**Cholesky Decomposition**

STATA saves the variance-covariance matrix from the underlying var in a variable called e(Sigma). Using this variable, e(Sigma),to calculate the Cholesky decomposition and interpret the results.

The first command names the e(Sigma) matrix as sig_var and the second command list the items in this matrix. The next command uses the function cholesky() to performa a cholesky decomposition of the sig_var matrix. Finally the last command displays the cholesky decomposition of the contemporaneous affect matrix.

**Interpretation of Consumption**

- Percentage increases in income have about twice contemporaneous affect that a percentage change in the value of investments.
- A 10% increase in investment value leads to a 2.5% increase in consumption in the current period
- A 10% increase in income leads to a 4% increase in in consumption in the current period

**Next Step-Impulse Response Functions
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Using the Cholesky decomposition the impact of changes in current income and investments were calculated for German consumption. What about the long-term affects on consumption from sudden unexpected increases in income? An unexpected gain in the stock market of 10% in 1 quarter may only boost consumption by 4% in the current quarter, but how about in later quarters? What is the response of consumption to sudden unexpected changes of other variables in the macroeconomy in the medium to long-term?

In order to attempt to answer these questions we would need to use the SVAR and Cholesky decomposition found in this post and calculate what are called Impulse Responses Functions. Impulse Response Functions are ideal for understanding how shocks to a system of equations, like a macroeconomic model, reverberate throughout the system across time.