A Time-Varying-Parameter Vector Autoregression Model with Stochastic Volatility – Part I

The basic Vector Autoregression (VAR) model is heavily used in macro-econometrics for explanatory purposes and forecasting purposes in trading. In recent years, a VAR model with time-varying parameters has been used to understand the interrelationships between macroeconomic variables. Since Primiceri (2005), econometricians have been applying these models using macroeconomic variables such as:

• Japan time series (Nakahima, 2011)
• US Bond yields (Fischer et al., 2022)
• Monthly Stock Indices from industrialized countries (Gupta et al., 2020)
• Peruvian exchange rate (Rodriguez et al., 2024)
• Indian exchange rate (Kumar, M., 2010)

This article extends the model usage to something our audience greatly cares about: trading! You’ll learn the basics of the estimation procedure and how to create a trading strategy based on the model.

Are you excited? I was when I started writing this article. Let me share what I’ve learned with you!

What is the difference between a basic VAR and a TVP-VAR-SV model?

All the explanations of the basic VAR can be found in our previous article. Here, we’ll provide the system of equations and compare them with our new model.

Let’s remember the basic model. For example, a basic bivariate VAR(1) can be described as a system of equations:

Or,

Where

A time-varying parameter VAR would be something like the following:

Do you get to see the difference between the two models? Not yet?

Let’s use matrices to see it clearly.

Where:

Now you see it?

The only difference is that the model’s parameters vary as time passes. Hence, it’s referred to as a  “time-varying-parameter” model.

Even though the difference appears simple,  the estimation procedure is much more complex than the basic VAR estimation.

You now say: I know we can have time-varying parameters, but where is the stochastic volatility in the previous equations?

Wait for it, my friend! We’ll see it later!

Don’t worry, we’ll keep it simple!

The TVP-VAR-SV model variables

The system of equations of the model

Using a new notation provided by Primiceri (2005):

Where:

Y: The vector of time series

B: The parameters of the lagged time series of this reduced model

A: The contemporary parameters of the time series vector

Sigma: The time-varying standard deviation (volatility) of each equation in the VAR.

Epsilon: A vector of shocks of each equation in the VAR.

What is the reduced model and what are contemporary parameters?

Well, in macroeconometrics, the reduced model can be understood as a simple VAR as modeled in our previous article. In this model, today’s time series values of the VAR vector are impacted only by their lag versions.

However, economists also talk about the impact that the same today’s time series values have on each other today’s time series values. This can be modeled as:

This can shown as a matrix below:

Which can also be presented as a system of equations:

The above model is understood in econometrics as a structural model to comprehend the time series interrelationships, contemporary or not, between the time series analyzed.

So, assuming we have daily data, the first question, which belongs to y1, has a12*y2 as today’s y2 impact on today’s y1. The same is true for the second question, which belongs to y2, where we see a21*y1, which is today’s time series y1 impact on y2. In a VAR, we have lag periods impacting today’s variables, in a structural VAR we have today’s variables impacting today’s other variables.

Due to these contemporary relationships, there is a problem called endogeneity, where the error terms epsilons are correlated with Y_t-1. To estimate a structural VAR, we need to clearly identify the matrix A variables. As Eric (2021) explained, there are 3 ways in the economic literature. But it’s not only that, as per this model, A is also time-varying. We’ll see later how this variables are estimated.

When you pre-multiply the system of equations by A^-1, you get something like:

Which can be further simplified as:

So,

Time-varying volatilities?

Yes! In a basic VAR, the error terms are homoskedastic, meaning, they present constant variance. In this case, we have variances that change over time; they’re time-variant.

The time-varying parameter stochastic behaviors

The basic VAR had its parameters constant. In this TVP-VAR-SV, we have almost all of our parameters time-variant. Due to this, we need to assign them stochastic processes. As in Primiceri (2005), we define them as:

We can then specify the matrix of variances of all the model’s shocks as:

Where I_n is the identity matrix and n is the number of time series in the VAR (in our case it’s 2). Q, S, and W are square positive-definite covariance matrices with a number of rows (or columns) equal to the number of parameters in B, A, and Sigma, respectively.

Something else to note: sigma is stochastic-based, which can be interpreted as stochastic volatility as, e.g., the Heston model is.

The priors

For a Bayesian inference, you always need priors. In the Primiceri (2005) algorithm, the priors are computed using your data sample’s first “T1” observations.

Using our previously defined variables, you can specify the priors (following Primiceri, 2005, and Del Negro and Primiceri, 2015):

Where

• N(): Normal distribution
• B_ols: This is the point estimate of the B parameters obtained by estimating a basic time-invariant VAR using the first T1 observations of the data sample.
• V(B_ols): This is the point estimate of the B parameters’ variances obtained by estimating a basic time-invariant structural VAR using the first T1 observations of the data sample. In B_0, the variance is multiplied by 4. This value can be named k_B.
• A_ols: This is the point estimate of the A parameters obtained by estimating a basic time-invariant structural VAR using the first T1 observations of the data sample.
• V(A_ols): This is the point estimate of the A parameters’ variances obtained by estimating a basic time-invariant structural VAR using the first T1 observations of the data sample. In A_0, this variance is multiplied by 4. This value can be named k_A.
• log(sigma_0): This is the point estimate of the standard errors obtained by estimating a basic time-invariant structural VAR using the first T1 observations of the data sample.}
• I_n: This is the identity matrix with “nxn” dimensions, where “n” is the number of time series used to estimate the VAR on them. Contrary to to A_0 and B_0, this variance is just multiplied by 1, where this value can be named k_sig.
• IW: The inverse Wishart distribution
• Q_0 follows an IW distribution with a scale matrix of k_Q^2 times 40 times V(B_ols) and 40 degrees of freedom
• W_0 follows an IW distribution with a scale matrix of k_W^2 times 2 times V(B_ols) and 2 degrees of freedom
• Q_0 follows an IW distribution with a scale matrix of k_S^2 times 2 times V(B_ols) and 2 degrees of freedom
• k_Q^2, k_W^2 and k_S^2 are 1, 0.01 and 0.1, respectively.

Once you estimate the priors with the first T1 observations, then you get the posterior distribution using the rest of the data sample.

The mixture of indicators

Before we dive into the algorithm, let’s learn something else. Do you remember the reduced-form model:

To clear the error term, we get

Primiceri (2005), appendix A.2 explains that the above model has a Gaussian non-linear state space representation. The difficulty with drawing Sigma_t is that they enter the model multiplicatively.

This presents the issue of not making it easy for the Kalman filter estimation done inside the whole estimation algorithm (The Kalman filter is linear-based). To overcome this issue, Primiceri (2005) applies squaring and takes the logarithms of every element of the previous equation. As a consequence of this transformation, the resulting state-space form becomes non-Gaussian, because the log(epsilon_t^2) has a log chi-squared distribution. To finally get a normal distribution for the error terms, Kim et al. (1998) use a mixture of normals to approximate each element of log(epsilon_t^2). Thus, the estimation algorithm uses the mixture indicators for each error term and each date.

Stay tuned for the next installment to read about the TVP-VAR-SV model estimation algorithm.

Originally posted on QuantInsti blog.

Go Source