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

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

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

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

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

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

Let’s bear in mind the fundamental mannequin. For instance, a fundamental bivariate VAR(1) might 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 can 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 fluctuate as time passes. Therefore, it’s known as a  “time-varying-parameter” mannequin.

Though the distinction seems easy,  the estimation process is rather more advanced than the fundamental 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?

Look forward to it, my pal! We’ll see it later!

Don’t fear, we’ll preserve 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 decreased 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 decreased mannequin and what are modern parameters?

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

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

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

This could proven as a matrix under:

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

Which can be introduced 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 each day knowledge, the primary query, which belongs to y1, has a12*y2 as at the moment’s y2 affect on at the moment’s y1. The identical is true for the second query, which belongs to y2, the place we see a21*y1, which is at the moment’s time collection y1 affect on y2. In a VAR, now we have lag intervals impacting at the moment’s variables, in a structural VAR now we have at the moment’s variables impacting at the moment’s different variables.

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

Whenever 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 might 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, that 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 fundamental VAR had its parameters fixed. On this TVP-VAR-SV, now we have virtually all of our parameters time-variant. Because 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 will 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 identification 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 plenty of 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 might be interpreted as stochastic volatility as, e.g., the Heston mannequin is.

The priors

For a Bayesian inference, you all the time 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 distributionB_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 information 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 information pattern. In B_0, the variance is multiplied by 4. This worth might 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 information 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 information pattern. In A_0, this variance is multiplied by 4. This worth might 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 information pattern.}I_n: That is the identification 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 might be named k_sig.IW: The inverse Wishart distributionQ_0 follows an IW distribution with a scale matrix of k_Q^2 instances 40 instances V(B_ols) and 40 levels of freedomW_0 follows an IW distribution with a scale matrix of k_W^2 instances 2 instances V(B_ols) and a pair of levels of freedomQ_0 follows an IW distribution with a scale matrix of k_S^2 instances 2 instances V(B_ols) and a pair of levels of freedomk_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 information 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 difficulty of not making it simple for the Kalman filter estimation finished 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 begin with, it’s worthwhile 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 obtained 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 and this different one to totally 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

Use the Kalman filter to replace the state equation and compute the probability.Pattern the variable from its posterior distribution utilizing a Metropolis-Hastings step.MCMC is Markov Chain Monte Carlo. Please discuss with our article to be taught extra about this kind of Monte Carlo and the Metropolis-Hastings algorithm.Theta is [B, A, V] the place these 3 variables had 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 is predicated 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 information and compute the log returns.

Let’s estimate the mannequin with all of the out there 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. It’s also possible to 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 depend 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 by way of least squares (LS). On this case, we set it to 1 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 intervals 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.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 really 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 may relate k_B, k_A and k_sig with the earlier part priors. Relating to the opposite inputs, see under:

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

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

Now we get to the place you wished! We’ll use the identical imported libraries and the identical dataframe referred to as var_data, which incorporates 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 match efficiency with our TVP-VAR-SV technique.The technique for each fashions might be lengthy solely.For the reason that TVP-VAR-SV mannequin estimation takes a number of time every buying and selling interval, now we have made the code script in order that if it’s worthwhile to cease the code from operating, you are able to do it and run it later by operating 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 fundamental VAR and the TVP-VAR-SV mannequin in case you may have finished so beforehand.

Then, we

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

Subsequent, we create the basic-VAR-based technique alerts. The code follows our earlier article.

Subsequent, we create the TVP-VAR-SV mannequin alerts via the same loop. This time we set tau to 40. The enter might 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 worthwhile 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 means that everytime you need to proceed operating the for loop, you may simply run the entire code.

Subsequent, we compute the technique returns. Now we have 4 methods

A Purchase-and-Maintain strategyA basic-VAR-based strategyA TVP-VAR-SV-based strategyA technique based mostly on the TVP-VAR-SV mannequin makes its lengthy alerts if and provided that the Purchase-and-Maintain cumulative returns are increased than their 15-window easy transferring common.

See,

Focusing solely on fairness returns, the fundamental VAR performs the worst. The TVP-VAR-SV and the SMA-based TVP-VAR-SV technique carry out carefully to the Purchase-and-hold technique. Nonetheless, the latter performs the most effective in virtually all of the years. Let’s see their buying and selling abstract statistics.

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

The fundamental VAR performs the worst with respect to not solely returns but in addition fairness volatility. That is mirrored in poor outcomes for the Sharpe, Calmar, and Sortino ratios. The utmost drawdown can also be big.The TVP-VAR-SV performs barely higher with respect to the Purchase-and-Maintain technique.The SMA-based TVP-VAR-SV is the most effective 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 kind of good, too.

Notes concerning the TVP-VAR-SV technique

There are some issues we have to have in mind whereas growing a method based mostly on this mannequin:

We selected tau equal to 40 arbitrarily, which might be not sufficient. Selecting one other quantity would possible produce completely different outcomes. The seed can also be arbitrarily chosen. Do a hyperparameter tuning to get the most effective outcomes whereas doing a walk-forward optimization.Now we have 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 rationale 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 knowledge samples, macroeconomic knowledge tends to be fitted with this mannequin in a short time despite the fact that they use nrep excessive. Because 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 at liberty to vary nrep at your comfort. Simply be sure that, in case you commerce hourly, the mannequin estimation ought to take much less time than an hour for dwell buying and selling, in case you commerce each 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 achieve this to enhance your outcomes.

Conclusion

Now we have delved into the fundamental definition of a TVP-VAR-SV mannequin. We then defined just a little bit the mannequin estimation, and eventually we opted for a buying and selling technique backtesting loop script to check the mannequin efficiency.

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

Would you like extra fashions to be examined?

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

References

Cogley, T. and Sargent, T. J. (2005), “Drifts and Volatilities: Financial Insurance policies and Outcomes within the Publish 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) Common Bayesian time-varying parameter VARs for predicting authorities bond yields. J Appl Econom 38(1):69–87Giannone, 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 trade charges, Worldwide Journal of Financial Sciences and Utilized Analysis, ISSN 1791-3373, Kavala Institute of Know-how, Kavala, Vol. 3, Iss. 2, pp. 21-39Nakajima, Jouchi, (2011), Time-Various Parameter VAR Mannequin with Stochastic Volatility: An Overview of Methodology and Empirical Purposes, 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 trade 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.

Writer: José Carlos Gonzáles Tanaka

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