<p>We can use a simple counting argument. In general: the number of additional restrictions = number of unknowns in the structural model - number of estimates in the reduced form.</p>
<p>Consider the structural equation
<span class="math-container">$$
B y_t = \Gamma_0 + \Gamma_1 y_{t-1} + \varepsilon_t.
$$</span>
Let us count the number of unknowns.</p>
<ul>
<li>The matrix <span class="math-container">$B$</span> has <span class="math-container">$K^2$</span> elements. However, following the slides you posted <a href="https://kevin-kotze.gitlab.io/tsm/ts-11-slide/#18" rel="nofollow noreferrer">here</a> the diagonal elements of <span class="math-container">$B$</span> are known to equal 1. So in total <span class="math-container">$B$</span> has <span class="math-container">$K^2 - K$</span> unknowns.</li>
<li>The vector <span class="math-container">$\Gamma_0$</span> has <span class="math-container">$K$</span> unknowns</li>
<li>The vector <span class="math-container">$\Gamma_1$</span> has <span class="math-container">$K^2$</span> unknowns</li>
<li>The errors <span class="math-container">$\varepsilon_t$</span> have mean zero and according to the slides, we assume that the covariances between the disturbance terms are also zero. So the only unknown for the variance-covariance matrix here are the variances of which there are <span class="math-container">$K$</span>.</li>
</ul>
<p>Adding everything together we have <span class="math-container">$2 K^2 + K$</span> unknowns</p>
<p>Now consider the reduced form:
<span class="math-container">$$
y_t = A_0 + A_1 y_{t-1} + u_t.
$$</span>
Here <span class="math-container">$A_0 = B^{-1} \Gamma_0, A_1 = B^{-1} \Gamma_1$</span> and <span class="math-container">$u_t = B^{-1} \varepsilon_t$</span>.</p>
<p>Let us count the number of coefficients we can estimate.</p>
<ul>
<li>The vector <span class="math-container">$A_0$</span> has <span class="math-container">$K$</span> coefficients.</li>
<li>The matrix <span class="math-container">$A_1$</span> had <span class="math-container">$K^2$</span> coefficients.</li>
<li>From the error <span class="math-container">$u_t$</span> we can estimates its (symmetric) variance-covariance matrix, which has <span class="math-container">$K + K(K-1)/2$</span> terms.</li>
</ul>
<p>In total we have <span class="math-container">$K + K^2 + K + K(K-1)/2 = K^2 + 2K + K(K-1)/2$</span> estimates.</p>
<p>So taking the difference between the number of unknowns and the number of estimates, we obtain:
<span class="math-container">$$
2 K^2 + K - K^2 - 2K - K(K-1)/2 = \frac{K(K-1)}{2}.
$$</span></p>
<p>Note that this is an estimate. In particular, imposing <span class="math-container">$K(K-1)/2$</span> additional restrictions will (in general) be necessary to identify all structural coefficients, but it might not be sufficient.</p>

CC BY-SA 4.0
2024-11-25T08:50:10.603Z
2024-11-25T08:50:10.603Z
tdm
<p>I'm currently using <a href="https://kevin-kotze.gitlab.io/tsm/ts-11-slide/#18" rel="nofollow noreferrer"><em>Structural vector autoregressive models</em></a> by Kevin Kotzé to learn Vector Autoregression. One of the points that it makes is the following:</p>
<blockquote>
<p><strong>the number of restrictions that we need to impose is equivalent to the number of terms in the lower (or upper) triangle of the B matrix, which is <span class="math-container">$(K^2-K)/2$</span></strong></p>
</blockquote>
<p>where our model is the following VAR(1) (<span class="math-container">$y_t$</span> is a K-dimensional vector of variables):</p>
<p><span class="math-container">$B y_t = \Gamma_0 + \Gamma_1 y_{t-1} + \varepsilon_t$</span>, which leads to the following reduced-form VAR:</p>
<p><span class="math-container">$y_t = A_0 + A_1 y_{t-1} + u_t$</span></p>
<p>In the bivariate case, this seems to make sense as with <span class="math-container">$K=2$</span>, we can easily solve the system of equations and find all the structural VAR parameters. However, it's not totally obvious to me:</p>
<p>a) Why <span class="math-container">$(K^2-K)/2$</span> is the magic number that allows us to deduce the structural parameters - I understand it is equivalent to the upper triangle of B.</p>
<p>b) How we know that given this amount of restrictions, we can always solve for all the structural parameters. I calculated by hand and was able to convince myself for <span class="math-container">$K=2$</span>, but it's not immediately obvious that higher dimensions will follow similarly.</p>
<p>I'm wondering if there are any proofs or more rigorous illustrations as to why this result can be true for all <span class="math-container">$K$</span>.</p>

<macroeconomics><econometrics><statistics><time-series><svar>
SVAR: proof of the $(K^2-K)/2$ restrictions and identifiability?
stats_learner
# SVAR: proof of the $(K^2-K)/2$ restrictions and identifiability?

(Asked by: username_0 on 2024-08-15)

I'm currently using [*Structural vector autoregressive models*](https://kevin-kotze.gitlab.io/tsm/ts-11-slide/#18) by Kevin Kotzé to learn Vector Autoregression. One of the points that it makes is the following:

> **the number of restrictions that we need to impose is equivalent to the number of terms in the lower (or upper) triangle of the B matrix, which is $(K^2-K)/2$**

where our model is the following VAR(1) ($y\_t$ is a K-dimensional vector of variables):

$B y\_t = \Gamma\_0 + \Gamma\_1 y\_{t-1} + \varepsilon\_t$, which leads to the following reduced-form VAR:

$y\_t = A\_0 + A\_1 y\_{t-1} + u\_t$

In the bivariate case, this seems to make sense as with $K=2$, we can easily solve the system of equations and find all the structural VAR parameters. However, it's not totally obvious to me:

a) Why $(K^2-K)/2$ is the magic number that allows us to deduce the structural parameters - I understand it is equivalent to the upper triangle of B.

b) How we know that given this amount of restrictions, we can always solve for all the structural parameters. I calculated by hand and was able to convince myself for $K=2$, but it's not immediately obvious that higher dimensions will follow similarly.

I'm wondering if there are any proofs or more rigorous illustrations as to why this result can be true for all $K$.

**Tags:** macroeconomics, econometrics, statistics, time-series, svar


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## 1 answers

### Answer by username_1 on 2024-11-25. Score: 1

We can use a simple counting argument. In general: the number of additional restrictions = number of unknowns in the structural model - number of estimates in the reduced form.

Consider the structural equation
$$
B y\_t = \Gamma\_0 + \Gamma\_1 y\_{t-1} + \varepsilon\_t.
$$
Let us count the number of unknowns.

* The matrix $B$ has $K^2$ elements. However, following the slides you posted [here](https://kevin-kotze.gitlab.io/tsm/ts-11-slide/#18) the diagonal elements of $B$ are known to equal 1. So in total $B$ has $K^2 - K$ unknowns.
* The vector $\Gamma\_0$ has $K$ unknowns
* The vector $\Gamma\_1$ has $K^2$ unknowns
* The errors $\varepsilon\_t$ have mean zero and according to the slides, we assume that the covariances between the disturbance terms are also zero. So the only unknown for the variance-covariance matrix here are the variances of which there are $K$.

Adding everything together we have $2 K^2 + K$ unknowns

Now consider the reduced form:
$$
y\_t = A\_0 + A\_1 y\_{t-1} + u\_t.
$$
Here $A\_0 = B^{-1} \Gamma\_0, A\_1 = B^{-1} \Gamma\_1$ and $u\_t = B^{-1} \varepsilon\_t$.

Let us count the number of coefficients we can estimate.

* The vector $A\_0$ has $K$ coefficients.
* The matrix $A\_1$ had $K^2$ coefficients.
* From the error $u\_t$ we can estimates its (symmetric) variance-covariance matrix, which has $K + K(K-1)/2$ terms.

In total we have $K + K^2 + K + K(K-1)/2 = K^2 + 2K + K(K-1)/2$ estimates.

So taking the difference between the number of unknowns and the number of estimates, we obtain:
$$
2 K^2 + K - K^2 - 2K - K(K-1)/2 = \frac{K(K-1)}{2}.
$$

Note that this is an estimate. In particular, imposing $K(K-1)/2$ additional restrictions will (in general) be necessary to identify all structural coefficients, but it might not be sufficient.


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