A time-varying-parameter vector autoregression mannequin with stochastic volatility

The essential Vector Autoregression (VAR) mannequin is closely utilized in macro-econometrics for explanatory functions and forecasting functions in buying and selling. Lately, a VAR mannequin with time-varying parameters has been used to grasp the interrelationships between macroeconomic variables. Since Primiceri (2005), econometricians have been making use of these fashions utilizing macroeconomic variables equivalent to:

• Japan time collection (Nakahima, 2011)
• US Bond yields (Fischer et al., 2022)
• Month-to-month Inventory Indices from industrialized international locations (Gupta et al., 2020)
• Peruvian alternate charge (Rodriguez et al., 2024)
• Indian alternate charge (Kumar, M., 2010)

This text extends the mannequin utilization to one thing our viewers significantly cares about: buying and selling! You’ll be taught the fundamentals of the estimation process and methods to create a buying and selling technique primarily based on the mannequin.

Are you excited? I used to be once I began writing this text. Let me share what I’ve realized with you!

This weblog covers:

What’s the distinction between a fundamental VAR and a TVP-VAR-SV mannequin?

All the reasons of the essential VAR will be present in our earlier article on VAR fashions. Right here, we’ll present the system of equations and evaluate them with our new mannequin.

Let’s bear in mind the essential mannequin. For instance, a fundamental bivariate VAR(1) will be described as a system of equations:

[
Y_{1,t} = phi_{11} Y_{1,t-1} + phi_{12} Y_{2,t-1} + u_{1,t}
]
[
Y_{2,t} = phi_{21} Y_{1,t-1} + phi_{22} Y_{2,t-1} + u_{2,t}
]

Or,

[
Y_t = Phi Y_{t-1} + U_t
]

The place

[
Y_t = begin{bmatrix} Y_{1,t} Y_{2,t} end{bmatrix}
]
[
Phi_t = begin{bmatrix} phi_{11} & phi_{12} phi_{21} & phi_{22} end{bmatrix}
]
[
Y_{t-1} = begin{bmatrix} Y_{1,t-1} Y_{2,t-1} end{bmatrix}
]
[
U_t = begin{bmatrix} u_{1,t} u_{2,t} end{bmatrix}
]

A time-varying parameter VAR could be one thing like the next:

[
Y_{1,t} = phi_{11,t} Y_{1,t-1} + phi_{12,t} Y_{2,t-1} + epsilon_{1,t}
]
[
Y_{2,t} = phi_{21,t} Y_{1,t-1} + phi_{22,t} Y_{2,t-1} + epsilon_{2,t}
]

Do you get to see the distinction between the 2 fashions? Not but?

Let’s use matrices to see it clearly.

[
Y_t = Phi_t Y_{t-1} + U_t
]

The place:

[
Y_t = begin{bmatrix} Y_{1,t} Y_{2,t} end{bmatrix}
]
[
Phi_t = begin{bmatrix} phi_{11,t} & phi_{12,t} phi_{21,t} & phi_{22,t} end{bmatrix}
]
[
Y_{t-1} = begin{bmatrix} Y_{1,t-1} Y_{2,t-1} end{bmatrix}
]
[
mathcal{E}_t = begin{bmatrix} epsilon_{1,t} epsilon_{2,t} end{bmatrix}
]

Now you see it?

The one distinction is that the mannequin’s parameters range as time passes. Therefore, it’s known as a  “time-varying-parameter” mannequin.

Though the distinction seems easy,  the estimation process is far more advanced than the essential VAR estimation.

You now say: I do know we are able to have time-varying parameters, however the place is the stochastic volatility within the earlier equations?

Watch for it, my pal! We’ll see it later!

Don’t fear, we’ll hold it easy!

The TVP-VAR-SV mannequin variables

The system of equations of the mannequin

Utilizing a brand new notation offered by Primiceri (2005):

$$Y_t = B_t Y_{t-1} + A_t^{-1} Sigma_t^{-1} varepsilon_t$$

The place:

Y: The vector of time collection

B: The parameters of the lagged time collection of this lowered mannequin

A: The modern parameters of the time collection vector

Sigma: The time-varying commonplace deviation (volatility) of every equation within the VAR.

Epsilon: A vector of shocks of every equation within the VAR.

What’s the lowered mannequin and what are modern parameters?

Properly, in macroeconometrics, the lowered mannequin will be understood as a easy VAR as modeled in our earlier article on VAR fashions. On this mannequin, at this time’s time collection values of the VAR vector are impacted solely by their lag variations.

Nevertheless, economists additionally speak concerning the affect that the identical at this time’s time collection values have on one another at this time’s time collection values. This may be modeled as:

$$A_t Y_t = C_t Y_{t-1} + Sigma_t varepsilon_t$$

This may proven as a matrix beneath:

$$start{bmatrix}
a_{11,t} & a_{12,t}
a_{21,t} & a_{22,t}
finish{bmatrix}
start{bmatrix}
y_{1,t}
y_{2,t}
finish{bmatrix}
=
start{bmatrix}
c_{11,t} & c_{12,t}
c_{21,t} & c_{22,t}
finish{bmatrix}
start{bmatrix}
y_{1,t-1}
y_{2,t-1}
finish{bmatrix}
+
start{bmatrix}
sigma_{1,t} & 0
0 & sigma_{2,t}
finish{bmatrix}
start{bmatrix}
epsilon_{1,t}
epsilon_{2,t}
finish{bmatrix}$$

Which can be offered as a system of equations:

$$start{aligned}
a_{11,t}y_{1,t} + a_{12,t}y_{2,t} &= c_{11,t}y_{1,t-1} + c_{12,t}y_{2,t-1} + sigma_{1,t}epsilon_{1,t}
a_{21,t}y_{1,t} + a_{22,t}y_{2,t} &= c_{21,t}y_{1,t-1} + c_{22,t}y_{2,t-1} + sigma_{2,t}epsilon_{2,t}
finish{aligned}$$

The above mannequin is known in econometrics as a structural mannequin to understand the time collection interrelationships, modern or not, between the time collection analyzed.

So, assuming now we have day by day knowledge, the primary query, which belongs to y1, has a12*y2 as at this time’s y2 affect on at this time’s y1. The identical is true for the second query, which belongs to y2, the place we see a21*y1, which is at this time’s time collection y1 affect on y2. In a VAR, now we have lag durations impacting at this time’s variables, in a structural VAR now we have at this time’s variables impacting at this time’s different variables.

As a result of these modern relationships, there’s a drawback known as endogeneity, the place the error phrases epsilons are correlated with Y_t-1. To estimate a structural VAR, we have to clearly determine the matrix A variables. As Eric (2021) defined, there are 3 methods within the financial literature. But it surely’s not solely that, as per this mannequin, A can also be time-varying. We’ll see later how this variables are estimated.

While you pre-multiply the system of equations by A^-1, you get one thing like:

$$Y_t = A_t^{-1} C_t Y_{t-1} + A_t^{-1} Sigma_t varepsilon_t$$

Which will be additional simplified as:

$$Y_t = B_t Y_{t-1} + U_t$$

So,

$$B_t = Phi_t = A_t^{-1} C_t
U_t = A_t^{-1} Sigma_t mathcal{E}_t$$

Time-varying volatilities?

Sure! In a fundamental VAR, the error phrases are homoskedastic, which means, they current fixed variance. On this case, now we have variances that change over time; they’re time-variant.

The time-varying parameter stochastic behaviors

The essential VAR had its parameters fixed. On this TVP-VAR-SV, now we have nearly all of our parameters time-variant. As a result of this, we have to assign them stochastic processes. As in Primiceri (2005), we outline them as:

$$start{aligned}
B_t &= B_{t-1} + nu_t
a_t &= a_{t-1} + zeta_t
log sigma_t &= log sigma_{t-1} + eta_t
finish{aligned}$$

We are able to then specify the matrix of variances of all of the mannequin’s shocks as:

$$V = Var left{ start{bmatrix} epsilon_t nu_t zeta_t eta_t finish{bmatrix} proper} = start{bmatrix} I_n & 0 & 0 & 0 0 & Q & 0 & 0 0 & 0 & S & 0 0 & 0 & 0 & W finish{bmatrix}$$

The place I_n is the id matrix and n is the variety of time collection within the VAR (in our case it’s 2). Q, S, and W are sq. positive-definite covariance matrices with numerous rows (or columns) equal to the variety of parameters in B, A, and Sigma, respectively.

One thing else to notice: sigma is stochastic-based, which will be interpreted as stochastic volatility as, e.g., the Heston mannequin is.

The priors

For a Bayesian inference, you at all times want priors. Within the Primiceri (2005) algorithm, the priors are computed utilizing your knowledge pattern’s first “T1” observations.

Utilizing our beforehand outlined variables, you may specify the priors (following Primiceri, 2005, and Del Negro and Primiceri, 2015):

$$start{aligned}
B_0 &sim N(B_{OLS}, 4V(B_{OLS}))
A_0 &sim N(A_{OLS}, 4V(A_{OLS}))
log sigma_0 &sim N(log sigma_{OLS}, I_n)
Q_0 &sim IW(k_Q^2 cdot 40 cdot V(B_{OLS}), 40)
W_0 &sim IW(k_W^2 cdot 2 cdot I_n, 2)
S_0 &sim IW(k_S^2 cdot 2 cdot V(A_{OLS}), 2)
finish{aligned}$$

The place

• N(): Regular distribution
• B_ols: That is the purpose estimate of the B parameters obtained by estimating a fundamental time-invariant VAR utilizing the primary T1 observations of the info pattern.
• V(B_ols): That is the purpose estimate of the B parameters’ variances obtained by estimating a fundamental time-invariant structural VAR utilizing the primary T1 observations of the info pattern. In B_0, the variance is multiplied by 4. This worth will be named k_B.
• A_ols: That is the purpose estimate of the A parameters obtained by estimating a fundamental time-invariant structural VAR utilizing the primary T1 observations of the info pattern.
• V(A_ols): That is the purpose estimate of the A parameters’ variances obtained by estimating a fundamental time-invariant structural VAR utilizing the primary T1 observations of the info pattern. In A_0, this variance is multiplied by 4. This worth will be named k_A.
• log(sigma_0): That is the purpose estimate of the usual errors obtained by estimating a fundamental time-invariant structural VAR utilizing the primary T1 observations of the info pattern.}
• I_n: That is the id matrix with “nxn” dimensions, the place “n” is the variety of time collection used to estimate the VAR on them. Opposite to to A_0 and B_0, this variance is simply multiplied by 1, the place this worth will be named k_sig.
• IW: The inverse Wishart distribution
• Q_0 follows an IW distribution with a scale matrix of k_Q^2 instances 40 instances V(B_ols) and 40 levels of freedom
• W_0 follows an IW distribution with a scale matrix of k_W^2 instances 2 instances V(B_ols) and a couple of levels of freedom
• Q_0 follows an IW distribution with a scale matrix of k_S^2 instances 2 instances V(B_ols) and a couple of levels of freedom
• k_Q^2, k_W^2 and k_S^2 are 1, 0.01 and 0.1, respectively.

When you estimate the priors with the primary T1 observations, then you definitely get the posterior distribution utilizing the remainder of the info pattern.

The combination of indicators

Earlier than we dive into the algorithm, let’s be taught one thing else. Do you bear in mind the reduced-form mannequin:

$$Y_t = B_t Y_{t-1} + A_t^{-1} Sigma_t varepsilon_t$$

To clear the error time period, we get

$$A_t(Y_t – B_t Y_{t-1}) = A_t hat{y}_t = Sigma_t varepsilon_t$$

Primiceri (2005), appendix A.2 explains that the above mannequin has a Gaussian non-linear state house illustration. The issue with drawing Sigma_t is that they enter the mannequin multiplicatively.

This presents the problem of not making it simple for the Kalman filter estimation achieved inside the entire estimation algorithm (The Kalman filter is linear-based). To beat this difficulty, Primiceri (2005) applies squaring and takes the logarithms of each component of the earlier equation. As a consequence of this transformation, the ensuing state-space kind turns into non-Gaussian, as a result of the log(epsilon_t^2) has a log chi-squared distribution. To lastly get a traditional distribution for the error phrases, Kim et al. (1998) use a mix of normals to approximate every component of log(epsilon_t^2). Thus, the estimation algorithm makes use of the combination indicators for every error time period and every date.

$$S^T equiv {s_t}_{t=1}^T$$

The TVP-VAR-SV mannequin estimation algorithm

To start with, it’s good to know the TVP-VAR mannequin estimation defined right here follows the Primiceri (2005) methodology and Del Negro and Primiceri (2015).

This system makes use of the modified Bayesian-based Gibbs sampling algorithm offered by Cogley and Sargent (2005) to estimate the parameters.

Now you say: What? Is that Chinese language?

We’ve received it! Don’t fear! Let’s clarify the algorithm in easy phrases intimately. To let you already know, concerning the Bayesian estimation method, please discuss with this text on Bayesian Statistics In Finance and this different one on Foundations of Bayesian Inference to completely be taught extra about it.

Let’s clarify the algorithm. Following Del Negro and Primiceri (2015), the algorithm consists of the next loop:

( textual content{for every MCMC iteration:} )

( hspace{1em}textual content{- Draw } Sigma^T textual content{ from } p(Sigma^T | y^T, theta^T, s^T) )

( hspace{1em}textual content{- Draw } theta textual content{ from } p(theta | y^T, Sigma^T) )

( hspace{1em}textual content{- Draw } s^T textual content{ from } p(s^T | y^T, Sigma^T, theta) )

The place

1. Use the Kalman filter to replace the state equation and compute the probability.
2. Pattern the variable from its posterior distribution utilizing a Metropolis-Hastings step.

• MCMC is Markov Chain Monte Carlo. Please discuss with our article on Introduction To Monte Carlo Evaluation to be taught extra about any such Monte Carlo and the Metropolis-Hastings algorithm.
• Theta is [B, A, V] the place these 3 variables have been outlined beforehand.
• p(e|d) is the corresponding likelihood distribution of “e” given “d”.

You iterate till you make the distribution converge. Though we are saying the algorithm relies on MCMC and Metropolis-Hastings, Primiceri (2005) applies his personal specs for his TVP-VAR-SV mannequin.

A TVP-VAR-SV estimation in R

Let’s see how we are able to estimate the mannequin on this programming language. Let’s set up the corresponding libraries.

Then let’s import them

Let’s import the info and compute the log returns.

Let’s estimate the mannequin with all of the obtainable knowledge and forecast the next-day return. To get this forecast, we get attracts from the converged posterior distribution and we use the imply of all of the attracts to output a forecast level estimate. You too can use the median or another measure of central tendency (Giannone, Lenza, and Primiceri, 2015).

Output:

[0.015880450, 0.013688861, 0.014319192, 0.002445156, 0.005108312, 0.020364678, 0.015684312]

These returns’ indicators will rely upon every day’s estimation.

There are 4 inputs to debate:

• tau: is the size of the coaching pattern used for figuring out prior parameters through least squares (LS). On this case, we set it to at least one yr: 250 observations. So, if now we have “n” observations, we use the primary 250 observations to get the priors and the final “n-250” for mannequin estimation.
• nf: Variety of future time durations for which forecasts are computed. On this case, we’re within the next-day return.
• nrep: It’s the variety of MCMC attracts excluding the burn-in observations. We set it to 300. You possibly can learn extra about it in our weblog on Introduction To Monte Carlo Evaluation.
• nburn: The variety of MCMC attracts used to initialize the sampler used for convergence to get the posterior distribution. We set it to twenty. So, since now we have 300 attracts, we compute the posterior distributions with the final 280 attracts (300-20).

The operate truly has extra inputs, let’s see them along with their default values:

k_B = 4, k_A = 4, k_sig = 1, k_Q = 0.01, k_S = 0.1, k_W = 0.01,

pQ = NULL, pW = NULL, pS = NULL

You possibly can relate k_B, k_A and k_sig with the earlier part priors. Relating to the opposite inputs, see beneath:

$$start{aligned}
k_Q &= 0.01
k_S &= 0.1
k_W &= 0.01
p_Q &= 40
p_W &= 2
p_S &= 2
finish{aligned}$$

A buying and selling technique utilizing the TVP-VAR-SV mannequin in R

Now we get to the place you wished! We are going to use the identical imported libraries and the identical dataframe known as var_data, which comprises the shares’ worth log returns.

Some issues to say:

• We initialize the forecasts from 2019 onwards.
• We estimate utilizing 1500 observations as span.
• We additionally estimate a fundamental VAR to check efficiency with our TVP-VAR-SV technique.
• The technique for each fashions will likely be lengthy solely.
• Because the TVP-VAR-SV mannequin estimation takes lots of time every buying and selling interval, now we have made the code script in order that if it’s good to cease the code from working, you are able to do it and run it later by working the entire code once more.

Let’s first outline the operate that can permit us to import the dataframe of the forecast outcomes dataframe of the essential VAR and the TVP-VAR-SV mannequin in case you have got achieved so beforehand.

Then, we

1. Set the preliminary date to start out the forecasting course of.
2. Import the saved df_forecasts dataframe, in any other case, we create a brand new one with out the earlier operate.
3. Set the span as 1500.

Subsequent, we create the basic-VAR-based technique indicators. The code follows our earlier article on  Vector AutoRegression mannequin.

Subsequent, we create the TVP-VAR-SV mannequin indicators by means of an analogous loop. This time we set tau to 40. The enter will be chosen arbitrarily so long as you respect the proportions between nrep and nburn.

The estimation of the mannequin every day will take some minutes, thus, the entire loop will take a very long time. Watch out. In case it’s good to flip off your pc earlier than the loop runs, you may simply activate as soon as once more your pc and run the script as soon as once more. The code is written in such a manner that everytime you wish to proceed working the for loop, you may simply run the entire code.

Subsequent, we compute the technique returns. We’ve got 4 methods

• A Purchase-and-Maintain technique
• A basic-VAR-based technique
• A TVP-VAR-SV-based technique
• A method primarily based on the TVP-VAR-SV mannequin makes its lengthy indicators if and provided that the Purchase-and-Maintain cumulative returns are increased than their 15-window easy shifting common.

See, Focusing solely on fairness returns, the essential VAR performs the worst.

The TVP-VAR-SV and the SMA-based TVP-VAR-SV technique carry out intently to the Purchase-and-hold technique. Nevertheless, the latter performs the very best in nearly all of the years. Let’s see their buying and selling abstract statistics.

The equity-curve casual conclusion will be additional confirmed by the abstract statistics.

1. The essential VAR performs the worst with respect to not solely returns but additionally fairness volatility. That is mirrored in poor outcomes for the Sharpe, Calmar, and Sortino ratios. The utmost drawdown can also be big.
2. The TVP-VAR-SV performs barely higher with respect to the Purchase-and-Maintain technique.
3. The SMA-based TVP-VAR-SV is the very best performer. It has an elevated 80% fairness curve return with respect to the Purchase-and-Maintain technique and the opposite statistics are clearly higher. The Sortino ratio is sort of good, too.

Notes concerning the TVP-VAR-SV technique

There are some issues we have to bear in mind whereas growing a technique primarily based on this mannequin:

• We selected tau equal to 40 arbitrarily, which might be not sufficient. Selecting one other quantity would probably produce completely different outcomes. The seed can also be arbitrarily chosen. Do a hyperparameter tuning to get the very best outcomes whereas doing a walk-forward optimization.
• We’ve got chosen nrep equal to 300. That is fairly small in comparison with macroeconometric requirements, the place nrep will get to be 50,000 in some circumstances. The explanation econometricians use such a big quantity is that macroeconomic knowledge samples are often very small in comparison with monetary knowledge samples. Because of the low amount of information samples, macroeconomic knowledge tends to be fitted with this mannequin in a short time despite the fact that they use nrep excessive. As a result of our span being equal to 1500, if we used nrep equal to 50,000, the estimation for every day will certainly take hours and even days. That’s why we use solely 300 as nrep. Please be happy to vary nrep at your comfort. Simply ensure that, in the event you commerce hourly, the mannequin estimation ought to take much less time than an hour for dwell buying and selling, in the event you commerce day by day, the mannequin estimation ought to take lower than a day, and so forth.
• We haven’t integrated stop-loss and take-profit targets. Please accomplish that to enhance your outcomes.

Conclusion

We’ve got delved into the essential definition of a TVP-VAR-SV mannequin. We then defined somewhat bit the mannequin estimation, and eventually we opted for a buying and selling technique backtesting loop script to check the mannequin efficiency.

Do you wish to be taught the fundamentals of the monetary time collection evaluation? Don’t hesitate to be taught from our course Monetary Time Sequence Evaluation for Buying and selling.

Would you like extra fashions to be examined?

Don’t hesitate to observe our weblog, we’re at all times creating extra methods for you!

References

• Cogley, T. and Sargent, T. J. (2005), “Drifts and Volatilities: Financial Insurance policies and Outcomes within the Put up WWII U.S.,” Assessment of Financial Dynamics, 8(2), 262-302.
• Del Negro, M. and Primiceri, G., (2015), Time Various Structural Vector Autoregressions and Financial Coverage: A Corrigendum, The Assessment of Financial Research, 82, difficulty 4, p. 1342-1345.
• Eric (2021) “Understanding and Fixing the Structural Vector Autoregressive Identification Downside” in https://www.aptech.com/weblog/understanding-and-solving-the-structural-vector-autoregressive-identification-problem/, consulted on August 1st, 2024.
• Fischer MM, Hauzenberger N, Huber F, Pfarrhofer M (2023) Normal Bayesian time-varying parameter VARs for predicting authorities bond yields. J Appl Econom 38(1):69–87
• Giannone, Domenico, Lenza, Michele and Primiceri, Giorgio, (2015), Prior Choice for Vector Autoregressions, The Assessment of Economics and Statistics, 97, difficulty 2, p. 436-451.
• Gupta, R., Huber, F., and Piribauer, P. (2020) Predicting worldwide fairness returns: Proof from time-varying parameter vector autoregressive fashions, Worldwide Assessment of Monetary Evaluation, Quantity 68, 101456, ISSN 1057-5219.
• Kim, S., Shephard, N., and Chib, S., (1998), Stochastic Volatility: Chance Inference and Comparability with ARCH Fashions, The Assessment of Financial Research, 65, difficulty 3, p. 361-393.
• Kumar, M., (2010) A time-varying parameter vector autoregression mannequin for forecasting rising market alternate charges, Worldwide Journal of Financial Sciences and Utilized Analysis, ISSN 1791-3373, Kavala Institute of Know-how, Kavala, Vol. 3, Iss. 2, pp. 21-39
• Nakajima, Jouchi, (2011), Time-Various Parameter VAR Mannequin with Stochastic Volatility: An Overview of Methodology and Empirical Functions, Financial and Financial Research, 29, difficulty, p. 107-142.
• Primiceri, Giorgio, (2005), Time Various Structural Vector Autoregressions and Financial Coverage, The Assessment of Financial Research, 72, difficulty 3, p. 821-852.
• Rodriguez, G., Castillo, P., Calero, R., Salcedo, R., Arellano, M. A., (2024), Evolution of the alternate charge pass-through into costs in Peru: An empirical software utilizing TVP-VAR-SV fashions, Journal of Worldwide Cash and Finance, Quantity 142, 2024, 103023, ISSN 0261-5606.

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