<p>I see that $n=2$ and $\mathbf \mu = \mathbb0$. 
At single (2-dimensional) observation level, we usually write (ignoring the disturbance vector)</p>

<p>$$\mathbf y_t = \Phi \mathbf y_{t,-1}$$</p>

<p>where dimensions are for your case $(2 \times 1) = (2 \times 2) \times (2 \times 1)$.</p>

<p>But notice that the standard regression formulation has the coefficient vector to be "to the right" of the regressor matrix... so in order to be able to apply the standard OLS estimation formulas, we transpose and obtain</p>

<p>$$\mathbf y'_t = \mathbf y_{t,-1}'\Phi'$$</p>

<p>and at full sample level we get</p>

<p>$$\mathbf Y' = \mathbf Y_{-1}'\Phi'$$</p>

<p>where with $T+1$ 2-dimensional observations available, the dimensions are ($T \times 2$) = $(T \times 2) \times  (2 \times 2)$</p>

<p>So, when applying the standard OLS formulas as your code does, the estimator estimates 
$\Phi'$, not $\Phi$.</p>

CC BY-SA 3.0
2017-04-28T13:49:37.030Z
Well spotted! I couldn't see the problem.
luchonacho
CC BY-SA 3.0
2017-04-26T15:38:39.407Z
2017-04-26T17:31:41.307Z
2017-04-26T17:31:41.307Z
Alecos Papadopoulos
<p>Model: $y_t = \Phi y_{t-1} + (\mathbb{I_n}-\Phi)\mu + \Sigma \varepsilon_t$
s.t. $y_t$ is a $(n \times 1)$ vector, and $\varepsilon_t \sim N(\vec{0},\mathbb{I}_n)$. $\mathbb{I}$ is the identity matrix. $\Phi$ and $\Sigma$ are $(n \times n)$ matrices.</p>

<p>Consider the following MATLAB code.</p>

<pre><code>Sigma = eye(2); %sqrtm(Covariance matrix)
Phi{1} = [0.9,0.12;0,0.9]; %Autoregressive matrix
mu = [0;0]; %Constant vector
nTs = 1000; %Length of simulation

%Specify Model
mdl = varm(2,1); 
mdl.AR = Phi;
mdl.Constant = mu;
mdl.Covariance = Sigma^2;

%Simulate Model
simTs = simulate(mdl,nTs); %Simulate VAR(1) (nTs x 2)

%Estimate Model
X = simTs(1:end-1,:);
Y = simTs(2:end,:); 
hatPhi = (X'*X)\(X'*Y)

&gt;&gt; hatPhi =
       0.8939  -0.0057
       0.1721   0.8732
</code></pre>

<p>Notice the output is the transpose of the AR matrix supplied. Why is this? I feel dumb, lol.</p>

<macroeconomics><econometrics><time-series><applied-econometrics>
When estimating VAR(1) by OLS, I get transpose of AR matrix. Why?
hipHopMetropolisHastings
# When estimating VAR(1) by OLS, I get transpose of AR matrix. Why?

(Asked by: username_0 on 2017-04-25)

Model: $y_t = \Phi y_{t-1} + (\mathbb{I_n}-\Phi)\mu + \Sigma \varepsilon_t$
s.t. $y_t$ is a $(n \times 1)$ vector, and $\varepsilon_t \sim N(\vec{0},\mathbb{I}_n)$. $\mathbb{I}$ is the identity matrix. $\Phi$ and $\Sigma$ are $(n \times n)$ matrices.
Consider the following MATLAB code.
Sigma = eye(2); %sqrtm(Covariance matrix)
Phi{1} = [0.9,0.12;0,0.9]; %Autoregressive matrix
mu = [0;0]; %Constant vector
nTs = 1000; %Length of simulation

%Specify Model
mdl = varm(2,1); 
mdl.AR = Phi;
mdl.Constant = mu;
mdl.Covariance = Sigma^2;

%Simulate Model
simTs = simulate(mdl,nTs); %Simulate VAR(1) (nTs x 2)

%Estimate Model
X = simTs(1:end-1,:);
Y = simTs(2:end,:); 
hatPhi = (X'*X)\(X'*Y)

>> hatPhi =
       0.8939  -0.0057
       0.1721   0.8732

Notice the output is the transpose of the AR matrix supplied. Why is this? I feel dumb, lol.

**Tags:** macroeconomics, econometrics, time-series, applied-econometrics

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## 1 answers

### Answer by username_1 on 2017-04-26. Score: 2 | Accepted answer

I see that $n=2$ and $\mathbf \mu = \mathbb0$.
At single (2-dimensional) observation level, we usually write (ignoring the disturbance vector)

$$\mathbf y\_t = \Phi \mathbf y\_{t,-1}$$

where dimensions are for your case $(2 \times 1) = (2 \times 2) \times (2 \times 1)$.

But notice that the standard regression formulation has the coefficient vector to be "to the right" of the regressor matrix... so in order to be able to apply the standard OLS estimation formulas, we transpose and obtain

$$\mathbf y'\_t = \mathbf y\_{t,-1}'\Phi'$$

and at full sample level we get

$$\mathbf Y' = \mathbf Y\_{-1}'\Phi'$$

where with $T+1$ 2-dimensional observations available, the dimensions are ($T \times 2$) = $(T \times 2) \times (2 \times 2)$

So, when applying the standard OLS formulas as your code does, the estimator estimates
$\Phi'$, not $\Phi$.

> **Comments:**
> **username_2**: Well spotted! I couldn't see the problem.  

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