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I have a simple model that I wish to fit using data. The model is of the form below.

\begin{gather} y_t = -\lambda r_t + \theta a_t + \varepsilon_1 \\ \\ \pi_t = \pi_{t-1} + w y_t + \varepsilon_2 \\ \\ a_t = -\alpha r_t + \varepsilon_3 \\ \\ r_t = \phi_1 \pi_1 + \phi_2 a_t + \varepsilon_4 \end{gather}

where $\varepsilon_i$ are i.i.d. Gaussian shocks. This model has matrix form:

\begin{gather} \begin{bmatrix} y_t \\ \pi_t \\ a_t \\ r_t \end{bmatrix} = \begin{bmatrix} 0 & 0 & \theta & -\lambda \\ w & 0 & 0 & 0 \\ 0 & 0 & 0 & -\alpha \\ 0 & \phi_1 & \phi_2 & 0 \end{bmatrix} \begin{bmatrix} y_t \\ \pi_t \\ a_t \\ r_t \end{bmatrix} + \begin{bmatrix} 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} y_{t-1} \\ \pi_{t-1} \\ a_{t-1} \\ r_{t-1} \end{bmatrix} + \begin{bmatrix} \varepsilon_1 \\ \varepsilon_2 \\ \varepsilon_3 \\ \varepsilon_4 \end{bmatrix} \end{gather}

i.e.

$$ Y_t = A Y_t + B Y_{t-1} + E $$

For estimating this model, assume we have a $4 \times N$ matrix of time series data for $y_t, \pi_t, a_t, r_t$ (call it $D$).

How do I fit this model using maximum likelihood estimation? (without using Kalman filters or Bayesian methods)

My understanding is that this is an "Errors-in-variables" problem, which in some cases requires Total Least Squares (TLS). However, the TLS methods I've seen apply to more simple scaler $y=\beta x + \gamma$ problems, rather than VARs (or structural VARs).

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  • $\begingroup$ If it's a DSGE model , then you're probably better off sending your question to economics.stackexchange.com. Also, if you google "horse race DSGE models" a lot of related papers will come up and some may be useful. The reason I googled for that was that there was a paper that surveyed the various methods for estimating DSGE models that had the phrase "horse race" in the title. I can't recall the exact name and it didn't come up in my google attempt but some other things did. It may be in the reference section of some of the papers that did come up since they too use the term horse race. $\endgroup$
    – mlofton
    Feb 8, 2022 at 1:37
  • $\begingroup$ Thanks, but I'm trying to stick with simple MLE methods for this small example, if at all possible... I believe Errors-in-variables is the right approach, but I'm unsure how to proceed with it. $\endgroup$
    – Mich55
    Feb 8, 2022 at 16:40
  • $\begingroup$ Hi Mich55. Unfortunately, I'm not knowledgeable enough about DSGE models to know if there is a simple approach. I just googled for "solving dsge models using error in variables" and a paper from NBER came up titled: "solution and estimation methods for DSGE models". The first author is Jose-Fernandez Villaverde. That may tell you if it's possible to go in your direction. Good luck. Oh, like I said, there may be people who could answer your question on economics.stackexchange.com. $\endgroup$
    – mlofton
    Feb 9, 2022 at 17:34
  • $\begingroup$ See also economics.stackexchange.com/questions/50345/… $\endgroup$ Feb 10, 2022 at 10:54

1 Answer 1

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I'm not particularly familiar with DSGE models, but a standard maximum likelihood estimation would go like this:

first we can write your matrix equation as:

$$ Y_t = (1-A)^{-1}BY_{t-1} + (1-A)^{-1}E$$

such that we can be express the conditional distribution of $Y_t$ given $Y_{t-1}$ :

$$ Y_t | Y_{t-1} \sim \mathcal N ( \hat Y_t , \Sigma )$$

where $ \hat Y_t = (1-A)^{-1}BY_{t-1}$, and $\Sigma = \sigma^2 (1 - A)^{-1}(1 - A^T)^{-1} $ (assuming $\varepsilon \sim \mathcal N (0,\sigma^2)$

This follows from standard properties of the multivariate normal distribution. The log-likelihood is therefore:

$$ \log\mathcal{L} = -\frac{1}{2}\sum_t (Y_t - \hat Y_t)^T\Sigma^{-1}(Y_t - \hat Y_t) - \frac{1}{2} N \log |\Sigma| $$

Now we can use the fact that $\Sigma^{-1} = (1 - A^T)(1 - A)/\sigma^2 $ to simplify some of the terms :

$$ \hat Y_t^T \Sigma^{-1} Y_t = Y_{t-1}^TB^T(1-A)Y_t = \pi_{t-1}\pi_t - \omega y_t y_{t-1}$$ $$ \hat Y_t^T \Sigma^{-1} \hat Y_t^T = Y_{t-1}B^TBY_{t-1} = y_{t-1}^2$$

Notice that we only have up to quadratic terms of $A$ from the exponential, so taking the derivative with respect to elements of $A$ from that part gives us something linear in $A$, e.g. :

$$ \frac{\partial}{\partial \theta} (\frac{1}{2} Y_t^T \Sigma^{-1} Y_t) = -\frac{1}{2\sigma^2} Y_t^T \left(\frac{dA^T}{d\theta}(1-A) + (1-A^T) \frac{dA}{d\theta}\right) Y_t$$

$$ = -\frac{1}{\sigma^2} \text{tr}\left( \frac{dA^T}{d\theta}(1-A) Y_t Y_t^T\right)$$

Notice that $\frac{dA^T}{d\theta}$ is a matrix that has just a single non zero entry at the location of $\theta$ , so we can write this more generally in a matrix notation

$$ \frac{\partial}{\partial A} (\frac{1}{2}\sum_t Y_t^T \Sigma^{-1} Y_t) = -\frac{1}{\sigma^2} (1 - A) \left(\sum_t Y_t Y_t^T \right) $$ (However keep in mind that this should be read as as equation for the non zero entries of $A$ only). Now we are left to deal with the determinant part, for which we can take the derivative as follows :

$$ \frac{\partial}{\partial \theta} \log |\Sigma| = -\text{tr}(\frac{\partial \Sigma^{-1}}{\partial \theta} \Sigma) $$

plugging in the derivative of $\Sigma^{-1}$ we get

$$ \frac{\partial}{\partial \theta} \log |\Sigma| = \frac{1}{\sigma^2}\text{tr}\left(\frac{dA^T}{d\theta}(1-A^T)^{-1} + (1-A)^{-1} \frac{dA}{d\theta}\right) = \frac{2}{\sigma^2}\text{tr}\left( \frac{dA^T}{d\theta}(1-A^T)^{-1} \right) $$

and as before we can write it as

$$ \frac{\partial}{\partial A} \log |\Sigma| = \frac{2}{\sigma^2}(1-A^T)^{-1} $$

putting everything together :

$$ \sigma^2 \frac{\partial}{\partial A} \log \mathcal{L} = (1 - A) \left(\sum_t Y_t Y_t^T \right) - B \left(\sum_t Y_t Y_{t-1}^T \right) - N(1-A^T)^{-1}.$$

Equating this to zero gives the maximum likelihood equation for $A$, But the equation is non linear and therefore can't be solved directly (in fact it can be written as a quadratic, but I don't think that helps much).

Nevertheless it can be easily solved iteratively using gradient descent methods, with the above gradient. Note that the dependence on the data $Y$ is only via the two correlation matrices defined above, which you only have to calculate once. Also note that the MLE does not depend on the noise variance $\sigma^2$.

[Note: I can't guarantee that every (-) sign and $^T$ is correct, if you are going to use some gradient based optimization tool you should verify the result against a numerical gradient, or you could use a symbolic program to calculate the derivatives. If you want to work out the details of the derivation by yourself, the Matrix Cookbook is a useful reference.]

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  • $\begingroup$ Thanks! Question: if the shocks have different variances (and/or non-zero covariance), is the result similar as long as you substitute $/sigma$ with $/Sigma$, and take care to keep it in the correct positions (since it may not commute) $\endgroup$
    – Mich55
    Mar 3, 2022 at 17:37
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    $\begingroup$ Yes that's correct. I'm not entirely sure if it will still factor out as in the simple case, but it should be easy to calculate as long you carefully take care of the orderings (if it does though then the MLE will remain independent of it) $\endgroup$
    – J. Delaney
    Mar 6, 2022 at 10:54
  • $\begingroup$ Do things change a lot if you don't know the covariance matrix (i.e. \Sigma), and have to estimate it? e.g., does it add an extra term to the likelihood function? $\endgroup$
    – Mich55
    Mar 6, 2022 at 23:21

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