What is the difference between SVAR and VAR

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by DM

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Learn things especially in coding need a practice. Lets put it into work then. Today we are going to enthusiatly dig deeper between what is VAR and what is structural VAR. 

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PhD is when you are trying to explain something and get so enthusiatly love about it.

The difference between VAR and SVAR 

VAR or vector autoregression 

Var or vector autoregression or unstructured vector autoregression is a way of learning the relationship between one variable without any structural limitation such as time. However, the problem with VAR only is that sometimes, there is no particular rule on seeing both two data related to each other. For example, in monetary policy, let's say we want to control the inflation of the price; as we expect that inflation will rise, the monetary authority will increase its instrument, the interest rate, and expect the commodity price to go down. It turns out that the price is not going down and rising instead. Then without any rule in concluding, opening the tide of economic mystery, we will complete that raising the interest rate will also increase the inflation or commodity price. 

Learn in the video below. 

SVAR or Structural vector autoregression

Meanwhile, Structural vector autoregression is similar to vector autoregression but with much more constraint—for example, time. One of the differences is in the sample. For example, the government attempted to reduce inflation by increasing the interest rate. In the unstructured vector autoregression, when the result of increasing the interest rate makes the inflation go up, we just read it as it is, which will create an erroneous conclusion. Meanwhile, in structure vector autoregression, we have a time duration that can help explain when the price increases and in which period the inflation goes down or starts to take effect, and whether it's significant. 

Some explanations in the video format

Here I found a couple of great videos about SVAR 

1. Why SVAR

Svar is a fascinating development in macroeconomics from Christopher Sims. From this video, for example, when the bank anticipates inflation while buying release more money, the inflation still rises. The wrong conclusion is that the interest rate hike led to inflation. But it turns out the monetary policy is an endogenous reaction to expected inflation.  The same issue occurred with fiscal policy; for example, we expect that there will be a reduction in private demand and, therefore, an increase in public spending, and the output will still decline. The wrong conclusion will be public spending will cause the work to fall. However, fiscal policy reacted endogenously with the reduction in production.  To measure the effect of policy, we need to identify or isolate purely exogenous independent movements and how the economy reacts to them. It calls impulse reaction.&nbsp Therefore we need to identify the structural model that isolates the exogenous variable from the model. After the economy is hit by the shock, getting the structural model is called identification.  According to Sims (1986) Identification is the interpretation of historically observed variation in data in a way that allows the variation to be used to predict the consequences of an action not yet undertaken.  After the structure is identified, one can predict inflation and output growth. And we can expect the fiscal effect on GDP. 

2. The calculation behind SVAR

ok 

How to identify purely exogenous shocl 

Lets say we have 

\[ AX_t = \beta_0+\beta X_{t-1} + u_t \] 

From the explanation above it can be interpreted that the vector \( X \) relies on its lag itself and structural shock \[ u \]. 

if we assume that X has 2 variables which is GDP gap y and interest rate r: it become

\[ X_t = \begin{bmatrix} y \\ r\end{bmatrix} \]

and make the system become

\[ \begin{matrix} y_t + a_{12} r_t = \beta_{10}+\beta_{11} y_{t-1}+\beta_{12} r_{t-1}+ u_{yt} \\a_{21} y_t +  r_t = \beta_{20}+\beta_{21} y_{t-1}+\beta_{22}r_{t-1}+ u_{rt} \end{matrix} \]

Where it can be written in matrix form

\[ \begin{bmatrix} 1 & a_{12} \\ a_{21} & 1 \end{bmatrix} \begin{bmatrix} y_t \\ r_t \end{bmatrix} = \begin{bmatrix} \beta_{10} \\ \beta_{20} \end{bmatrix} \begin{bmatrix} \beta_{11} & \beta_{12} \\ \beta_{21} & \beta_{22}\end{bmatrix} \begin{bmatrix} y_{t-1} \\ r_{t-1} \end{bmatrix} +\begin{bmatrix} u_{yt} \\ u_{rt} \end{bmatrix} \]

where 

\[ A = \begin{bmatrix} 1 & a_{12} \\ a_{21} & 1 \end{bmatrix} \]

What is identity matrices, check this video also 

And also what is inverse matrices

How to find inverse matrix

Test the matrix 

In Matrix A, the constants 1 and 1 showing a contemporaneous relationship.

\[ \begin{matrix} 1 & 2 & 3 \\ a & b & c \end{matrix} \]

test the matrix 

\[ \begin{bmatrix} 1 & 2 & 3 \\ a & b & c \end{bmatrix} \]

\[ \begin{bmatrix} 1 & 2 & 3 \\ a & b & c \end{bmatrix} \]

3. 

Link of SVAR calculation

There are several links to SVAR calculation that can help us understand deeper 

  • SVAR in Matlab 
  • SVAR in R 
Calculation

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