ivxPredictive

Unified Inference with General Autoregressive Roots

Description

The R package 'ivxPredictive' (under development) implements robust econometric inference methodologies for predictive regression models with autoregressive roots across the spectrum of stationarity and nonstationarity, with persistence types as defined by Magdalinos and Phillips (2020). In particular, the 'ivxPredictive' package extends the original IVX instrumentation of Phillips and Magdalinos (2009) and Kostakis, Magdalinos and Stamatogiannis (2015), to general autoregressive roots for parametric functional forms. Specifically, here we focus on linear specifications for the predictive regression model and implement the proposed procedure for the case of a conditional mean loss function (see, Magdalinos and Petrova (2022)) as well as a conditional quantile loss function (see, Katsouris, C. (2022)).

We consider the following persistence classes:

(P1). nearly stable processes:

$$\text{if} \ \ \left( \theta_n \right)_{ n \in \mathbb{N} } \ \ \text{is such that} \ \zeta = - \infty \ \ \text{and it holds that} \ \ \theta_n \to | \theta | < 1.$$

(P2). nearly unstable processes:

$$\text{if} \ \ \left( \theta_n \right)_{ n \in \mathbb{N} } \ \ \text{is such that} \ \ \zeta \equiv c \in \mathbb{R} \ \ \text{and it holds that} \ \ \theta_n \to \theta = 1.$$

(P3). nearly explosive processes:

$$\text{if} \ \ \left( \theta_n \right)_{ n \in \mathbb{N} } \ \ \text{is such that} \ \ \zeta = + \infty \ \ \text{and it holds that} \ \ \theta_n \to | \theta | > 1.$$

Methodology

This R package implements a novel endogenous instrumentation approach based on the IVX estimator examined by Phillips and Magdalinos (2009) and Kostakis, Magdalinos and Stamatogiannis (2015). The current procedure has a similar construction to the IV instrumentation proposed in the recent working paper of Magdalinos and Petrova (2022), with the aim to provide uniform and robust to the nuisance parameter of persistence inference across the spectrum of stationary and nonstationary roots, specifically for quantile autoregressive processes. We call this variant of the original IVX estimator, IVX-P, which can be employed to both conditional mean and conditional quantile functional forms when the model includes either univariate or multivariate regressors. The novelty of the IVX-P estimator is that is a 'hybrid estimator' which in contrast to the classical least squares estimator has desirable asymptotic theory properties and is constructed based on the underline nonstationary stochastic processes using information both within the admissible parameter space as well as outside the usual parameter space.

Installation (under development)

The R package 'ivxPredictive' will be able to be installed from Github.

Usage

# After development the package will be able to be installed using
install.packages("ivxPredictive")
library("ivxPredictive")

Notes:

1. We provide an open source R package ‘ivxPredictive’ that contains the implementation of both our IVX-P M-estimator together with inference tools for predictive regression models with persistence properties across the spectrum of stationary and nonstationary roots.

2. The implementation of the original (current) IVX instrumentation method proposed by Phillips and Magdalinos (2009) along with statistical inference procedures and empirical applications can be found in the R package 'ivx' developed by Kostas Vasilopoulos.

3. The Matlab code of the original IVX instrumentation method can be found on the website of Michalis Stamatogiannis.

4. The author would like to thank Ji Hyung Lee for kindly sharing the Matlab code for the implementation of the original IVX instrumentation method for the quantile predictive regression.

Application I: Robust Estimation and Inference

Empirical Illustrations

Example 1: (Autocorrelation Matrix Estimation)

Consider the predictive regression model with multiple predictors. Then, the Aols vector of parameters (not a matrix in this case, since the response is a scalar vector) and the Rols matrix of coefficients can be obtained as below

lm1  <- lm(yt ~ Xlag)
Aols <- coefficients(lm1)  

Rn   <- matrix(0, p, p)
for (i in 1:p) 
{
   Rn[i, i] <- lm( Xt[, i] ~ 0 + Xlag[, i] )$coefficients
}
  
> Rn
          [,1]      [,2]       [,3]      [,4]     [,5]        [,6]     
[1,] 0.9891264 0.0000000 0.00000000 0.0000000 0.000000  0.00000000  
[2,] 0.0000000 0.8461566 0.00000000 0.0000000 0.000000  0.00000000  
[3,] 0.0000000 0.0000000 0.07760762 0.0000000 0.000000  0.00000000  
[4,] 0.0000000 0.0000000 0.00000000 0.9985889 0.000000  0.00000000  
[5,] 0.0000000 0.0000000 0.00000000 0.0000000 1.000366  0.00000000 
[6,] 0.0000000 0.0000000 0.00000000 0.0000000 0.000000  1.04066777  

Remarks:

(i) The $R_n$ which is a $(p \times p)$ matrix corresponds to the autocorrelation coefficients of the nonstationary regressors of the predictive regression model. In the theoretical speciciation of the predictive regression model, these are modeled using an AR(1) model with a local-unit-root expression to capture persistence and near the unity characteristics. From the empirical example above, we see that some of these coefficients are close to the unit boundary from below and some reach the unit boundary from the explosive side.

(ii) The autocorrelation matrix that corresponds to the autoregressive specification of regressors is estimated based on least-squares optimization. The OLS estimator of the autocorrelation matrix similar to the case of univariate autoregressions is not a consistent estimator when the local-to-unity specification is imposed. However, the particular estimator is required when estimating statistics from fitted predictive regression models.

(iii) Another important aspect when fitting predictive regression models and especially when constructing test statistics such as testing for parameter instability in predictive regressions is to apply correctly the demeaning of random variables. Specifically, when the true model is assumed to have a non-zero model intercept then demeaning the random variables of the model can alter the asymptotic theory of estimators and test statistics depending on the underline persistence properties of regressors.

Example 2: (Covariance Matrix Estimation)

The estimation procedure for the covariance matrix of estimators and test statistics in predictive regression models does not follow conventional formulations as in the case of stationary time series regression models. Due to the local-to-unity specification of the autocorrelation coefficient(s), which captures the unknown 'degree of persistence' in time series regressors, the predictive regression specification permits to include both 'nonstationary predictors' and/or 'cointegrated regressors'; therefore the estimation of the covariance matrix is obtained using nonparametric kernel methodologies in order to introduce an endogeneity bias correction.

# Step 1: Estimation of the Rn autocorrelation coefficient 
for i=1:l
  rn(i,i) = regress( xt(:,i), xlag(:,i) );
end

# autoregressive residual estimation 
u = xt-xlag*rn;

# residuals' correlation matrix
corrmat = corrcoef([epshat u]);

# covariance matrix estimation (predictive regression)
covepshat = epshat'*epshat/nn;
covu = zeros(l,l);

for t=1:nn
    covu = covu+u(t,:)'*u(t,:);
end

# covariance matrix estimation (autoregression)
covu=covu/nn;
covuhat=zeros(1,l);

for i=1:l
    covuhat(1,i)=sum(epshat.*u(:,i));
end

Remarks:

(i)

(ii)

Example 3: (Instrumental Variables Estimation)

Firstly, we consider estimation procedure for the original IVX instrumentation of Phillips and Magdalinos (2009) and Kostakis, Magdalinos and Stamatogiannis (2015) and discuss the main features. Secondly, we present the estimation procedure of the IVX-P estimator (i.e., the IVX instrumentation that covers the whole spectrum of persistence classes in a unified way) and discuss the main features.

# Estimation of the original IVX instruments 

Example 4: (Demeaning of variables)

An important aspect which affects both the asymptotic theory of estimators as well as empirical results is the inclusion of model intercept in the specification form of the nonstationary predictive regression system. In particular, the IVX estimator is found to be invariant to whether the constructed instrument $z_{t-1}$ is demeaned or not, which can be easily seen by considering the corresponding asymptotic terms.

However, the presence of a non-zero model interecept has an impact on the limiting distribution of test statistics and estimators, thus a commonly used approach is to consider the case of a constant versus a zero intercept separately.

Explosive Behaviour Illustrative Examples:

Notice that various Software packages now exist which can be employed for monitoring time series data for explosive bubbles, although the main aim of developing the R package 'ivxPredictive' is for estimation and inference purposes, we briefly mention some of them here. Specifically, the PWY (2011) and PSY (2015) statistical tests for monitoring explosive behaviour are implemented in MATLAB, R (package 'psymonitor') as well as in Eviews. Furthermore, the R package 'exuber' developed by Kostas Vasilopoulos provides the corresponding R implementation while in Stata a coding implementation is discussed by Baum and Otero, 2021. Lastly, a review on the current literature of testing for explosive bubbles can be also found in this study literature.

Illustrative Example 1

// installing the R package exuber
install.packages("exuber")
library(exuber)

> rsim_data

-- radf (minw = 19, lag = 0) ------------------------------------

     id     adf   sadf   gsadf
   psy1  -2.461  1.946   5.190
   psy2  -2.858  7.880   7.880
  evans  -5.830  5.283   5.985
    div  -1.950  1.113   1.335
   blan  -5.146  3.930  10.951

  gsadf_panel
        2.407

Illustrative Example 2

We begin with an example of the naive estimation of the quantile autoregressive model when the underline stochastic process has a mildly explosive or explosive behaviour but a stationary quantile autoregressive model is fitted to the data. This example, illustrates exactly that usefulness of developing a unified framework that ensures robust inference regardless the persistence properties of the time series. Although, the estimates below are likley to be biased, these large values of model estimates demonstrate the importance of the proposed IVX-P estimator which takes into account such explosive stochastic behaviour and ensures that both empirical as well as asymptotic theory results are valid.

mydata      <- read.table("crypto.txt", header = TRUE)
crypto_data <- as.matrix(mydata)

etherum <- as.matrix( as.vector( crypto_data[ ,2] ) )
bitcoin <- as.matrix( as.vector( crypto_data[ ,3] ) )

nr    <- nrow( etherum )
e_t   <- as.matrix( etherum[2:nr,1] )
e_lag <- as.matrix( etherum[1:(nr-1),1] )
b_t   <- as.matrix( bitcoin[2:nr,1] )
b_lag <- as.matrix( bitcoin[1:(nr-1),1] )

model.LM_etherum <- lm( e_t ~ e_lag )
summary( model.LM_etherum  )

model.LM_bitcoin <- lm( b_t ~ b_lag  )
summary( model.LM_bitcoin  )

tau <- 0.95
###### Model 1: Etherum 
model.QR_etherum <- rq( e_t ~ e_lag, tau = tau )
model.summary    <- summary( model.QR_etherum , se = "boot", bsmethod= "xy" )

###### Model 2: Bitcoin
model.QR_etherum <- rq( b_t  ~ b_lag, tau = tau )
model.summary    <- summary( model.QR_etherum , se = "boot", bsmethod= "xy" )

Illustrative Example 3

Application II: Forecasting

In particular, we consider a forecasting exerice in which a fixed-size moving window is employed to estimate out-of-sample forecasting sequences based on the predictive regression model and the (original) ivx instrumentation. Notice that two functions can be employed for the forecasting exercise: (i) "ivx_forecast" function: produces the y_hat_forecast based on the ivx instrumentation and (ii) "forecast_scheme" function: implements the fixed-size rolling window forecasting exercise.

ivx_forecast <- function( y.t.w = y.t.window, x.t.w = x.t.window, x.lag.w = x.lag.window, h = 1 ) 
{# begin of function
  
  y.t.w   <- y.t.w
  x.t.w   <- x.t.w
  x.lag.w <- x.lag.w
  
  y.t.w   <- as.matrix( y.t.w )
  x.t.w   <- as.matrix( x.t.w )
  x.lag.w <- as.matrix( x.lag.w )
  
  nn <- NROW( x.lag.w )
  f  <- NCOL( x.lag.w )
  
  # First stage regression i.e., yt on xt
  lm1  <- lm(y.t.w ~ x.lag.w - 1 )
  Aols <- coefficients(lm1)
  
  #epshat contains the residuals of the predictive regression
  epshat <- matrix( residuals(lm1) )
  
  Rn <- matrix(0, f, f)
  for (i in 1:f)
  {
    Rn[i, i] <- lm( x.t.w[, i] ~ 0 + x.lag.w[, i] )$coefficients
  }
  
  # autoregressive residual estimation 
  u <- x.t.w - x.lag.w %*% Rn
  
  # residuals' correlation matrix
  corrmat   <- cor(cbind(epshat, u))
  
  # covariance matrix estimation 
  covepshat <- crossprod(epshat) / nn
  covu      <- matrix(0, f, f)
  
  for (i in 1:nn)
  {
    covu <- covu + crossprod(u[i, , drop = FALSE])
  }
  
  covu    <- covu / nn
  covuhat <- matrix(0, 1, f)
  
  for (i in 1:f) 
  {
    covuhat[, i] <- sum(epshat * u[, i])
  }
  
  covuhat <- t(covuhat) / nn
  m       <- floor(nn^(1 / 3)) # bandwith parameter
  uu      <- matrix(0, f, f)
  
  for (i in 1:m) 
  {
    a <- matrix(0, f, f)
    for (j in (i + 1):nn) 
    {
      a <- a + t(u[j, , drop = F]) %*% u[j - i, , drop = F]
    }
    uu <- uu + (1 - i / (m + 1)) * a
  }
  uu <- uu / nn
  
  # Estimation of the Omegauu matrix 
  Omegauu <- covu + uu + t(uu)
  
  q <- matrix(0, m, f)
  for (i in 1:m) 
  {
    p <- matrix(0, nn - i, f)
    
    for (j in (i + 1):nn)
    {
      p[j - i, ] <- u[j, , drop = F] * epshat[j - i] # epshat should be transposed
    }
    q[i, ] <- (1 - i / (1 + m)) * colSums(p)
  }
  residue <- apply(q, 2, sum) / nn
  Omegaeu <- covuhat + residue # resideue should be transposed
  
  # instrument construction
  h <- 1
  n      <- nn - h + 1
  Rz     <- (1 - 1 / (nn^0.95)) * diag(f)
  diffx  <- x.t.w - x.lag.w
  z      <- matrix(0, nn, f)
  z[1, ] <- diffx[1, ]
  
  for (i in 2:nn) 
  {
    z[i, ] <- z[i - 1, ] %*% Rz + diffx[i, ]
  }
  
  Z  <- rbind(matrix(0, 1, f), z[1:(n - 1),  , drop = F])
  zz <- rbind(matrix(0, 1, f), z[1:(nn - 1), , drop = F])
  ZK <- matrix(0, n, f)
  
  # for computations below we just use the notation n for the sample size 
  for (i in 1:n) 
  {
    ZK[i, ] <- colSums(zz[i:(i + h - 1), , drop = F])
  }
  
  yy <- matrix(0, n, 1)
  # Practically yy and y.t are the same since h=1 
  for (i in 1:n)
  {
    yy[i] <- sum(y.t.w[i:(i + h - 1), drop = F])
  }
  
  # Practically x.lag and xk.lag are the same since h=1 
  xK <- matrix(0, n, f)
  for (i in 1:n)
  {
    xK[i, ] <- colSums(x.lag.w[i:(i + h - 1), , drop = F])
  }
  
  meanxK <- colMeans(xK)
  Yt     <- yy - mean(yy)
  Xt     <- matrix(0, n, f)
  
  for (i in 1:f)
  {
    Xt[, i] <- xK[, i, drop = F] - meanxK[i] * matrix(1, n, 1)
  }
  
  # Computation of the Aivx matrix 
  Aivx     <- t(Yt) %*% Z %*% pracma::pinv(t(Xt) %*% Z)
  
  xK.t <- matrix(0, n, f)
  for (i in 1:n)
  {
    xK.t[i, ] <- colSums(x.t.w[i:(i + h - 1), , drop = F])
  }
  
  meanxKplus1 <- colMeans(xK.t)
  Xtplus1     <- matrix(0, n, f)
  
  for (i in 1:f)
  {
    Xtplus1[, i] <- xK.t[, i, drop = F] - meanxKplus1[i] * matrix(1, n, 1)
  }
  
  forecast    <- Aivx%*%( as.matrix( Xtplus1[nn, ]) ) 
  Yt.new      <- rbind ( Yt, forecast )
  Yt.new.mean <- mean( Yt.new )
  yy.forecast <- Yt.new + Yt.new.mean
  
  nrows      <- NROW( yy.forecast )
  y.forecast <- as.numeric( yy.forecast[nrows ,1] )
  
  return( y.forecast )
  
}#end of function

Bibliography

Key References:

• Katsouris, C. (2022b). "Asymptotic Theory for Moderate Deviations from the Unit Boundary in Quantile Autoregression and Predictive Regression Models". University of Southampton, Working paper.
• Katsouris, C. (2022a). "Asymptotic Theory for Moderate Deviations from the Unit Boundary in Quantile Autoregressive Time Series". arXiv preprint arXiv:2204.02073.
• Katsouris, C. (2021d). "Testing for Structural Breaks in Predictive Regression Models". University of Southampton, Working paper.
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• Kostakis, A., Magdalinos, T., & Stamatogiannis, M. P. (2015). "Robust econometric inference for stock return predictability". The Review of Financial Studies, 28(5), 1506-1553.
• Lee, J. H. (2016). "Predictive quantile regression with persistent covariates: IVX-QR approach". Journal of Econometrics, 192(1), 105-118.
• Fan, R., & Lee, J. H. (2019). Predictive quantile regressions under persistence and conditional heteroskedasticity. Journal of Econometrics, 213(1), 261-280.
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• Magdalinos, T., & Petrova, K. (2022). "Uniform and distribution-free inference with general autoregressive processes". University of Southampton, Working paper.
• Magdalinos, T., & Phillips, P. C. B. (2020). "Econometric inference in matrix vicinities of unity and stationarity". University of Southampton, Working paper.
• Magdalinos, T., & Phillips, P. C. B. (2009). "Limit theory for cointegrated systems with moderately integrated and moderately explosive regressors". Econometric Theory, 25(2), 482-526.
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References on IVX Variants:

• Christou, C., Gupta, R., Hassapis, C., & Suleman, T. (2018). The role of economic uncertainty in forecasting exchange rate returns and realized volatility: Evidence from quantile predictive regressions. Journal of Forecasting, 37(7), 705-719.
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• Demetrescu, M., Georgiev, I., Rodrigues, P. M., & Taylor, A. R. (2022). Extensions to IVX methods of inference for return predictability. Journal of Econometrics.
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• Ren, Y., Tu, Y., & Yi, Y. (2019). Balanced predictive regressions. Journal of Empirical Finance, 54, 118-142.
• Xu, K. L. (2020). Testing for multiple-horizon predictability: Direct regression based versus implication based. The Review of Financial Studies, 33(9), 4403-4443.
• Yang, B., Long, W., & Yang, Z. (2022). Testing predictability of stock returns under possible bubbles (IVX-BUB). Journal of Empirical Finance.
• Yang, B., Long, W., Peng, L., & Cai, Z. (2020). Testing the predictability of US housing price index returns based on an IVX-AR model (IVX-AR). Journal of the American Statistical Association, 115(532), 1598-1619.

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1990's

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2000's

• Aue, A., & Horváth, L. (2007). A limit theorem for mildly explosive autoregression with stable errors. Econometric Theory, 23(2), 201-220.
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Remark:

In contrast to the ordinary least squares estimates, the opposite result seems to hold for the original IVX estimator, that is, it seems to not uniformly be improving when many parameters are estimated. The particular aspect, is currently an open question in the literature which worths further investigation. Some studies in this direction include:

• Lee, J. H., Shi, Z., & Gao, Z. (2022). On LASSO for predictive regression. Journal of Econometrics, 229(2), 322-349.
• Ke-Li Xu & Junjie Guo (2022). A New Test for Multiple Predictive Regression. Journal of Financial Econometrics.

Code of Coduct

Please note that the ‘ivxPredictive’ project will be released with a Contributor Code of Coduct (under construction). By contributing to this project, you agree to abide by its terms.

Declarations

The author (Christis G. Katsouris) declares no conflicts of interest.

In particular, the author declares that has no affiliations with or involvement in any organization or entity with any financial interest (such as honoraria; educational grants; participation in speakers’ bureaus; membership, employment, consultancies, stock ownership, or other equity interest; and expert testimony or patent-licensing arrangements), or non-financial interest (such as personal relationships) in the subject matter or materials discussed in the manuscript and implemented in the R package.

Acknowledgments

The author greatfully acknowledges financial support from Graduate Teaching Assistantships at the School of Economic, Social and Political Sciences of the University of Southampton as well as from the Vice-Chancellor's PhD Scholarship of the University of Southampton, for the duration of the academic years 2018 to 2021. The author also gratefully acknowledges previously obtained funding from Research Grants of interdisciplinary Centers of research excellence based at the University of Cyprus (UCY) as well as at the University College London (UCL).

Part of the aspects implemented in the R package 'ivxPredictive', are discussed in the PhD thesis of the author (Christis G. Katsouris) titled: "Aspects of Estimation and Inference for Predictive Regression Models", completed under the supervision of Professor Jose Olmo and Professor Anastasios Magdalinos. In his PhD thesis, Katsouris C. generalises the work of Phillips P.C.B (under the guidance of his advisor Magdalinos T.) on robust econometric inference for nonstationary time series models, under more general persistence conditions than his predecessors. Furthermore, he proposes testing methodologies for structural break detection in predictive regressions.

An Econometrician amongst Statisticians or A Statistician amongst Econometricians?

Christis G. Katsouris has participated to various international academic conferences such as: (i) the International Symposium on Nonparametric Statistics (ISNS), June 2022 in Paphos, Cyprus, (ii) the Hausdorff Center for Mathematics Summer School in High-Dimensional Statistics, July 2021 in Bonn, Germany and (iii) the International Association for Applied Econometrics, Annual Conference, June 2019 in Nicosia, Cyprus.

Historical Background

Standing on the shoulders of giants.

''If I have been able to see further, it was only because I stood on the shoulders of giants." Isaac Newton, 1676

Aleksandr Lyapunov

Aleksandr Lyapunov (6 June 1857 – 3 November 1918) was a Russian mathematician, mechanician and physicist. Lyapunov is known for his development of the stability theory of a dynamical system, as well as for his many contributions to mathematical physics and probability theory. A major theme in Lyapunov's research was the stability of a rotating fluid mass with possible astronomical application. His main preoccupations were the stability of equilibria and the motion of mechanical systems, especially rotating fluid masses, and the study of particles under the influence of gravity. Lyapunov's work in the field of mathematical physics regarded the boundary value problem of the equation of Laplace. In the theory of potential, his work from 1897: "On some questions connected with Dirichlet's problem" clarified several important aspects of the theory. His work in this field is in close connection with the work of Steklov. Lyapunov developed many important approximation methods. His methods, which he developed in 1899, make it possible to define the stability of sets of ordinary differential equations. In the theory of probability, he generalised the works of Chebyshev and Markov, and proved the Central Limit Theorem under more general conditions than his predecessors. The method of characteristic functions he used for the proof later found widespread use in probability theory (Source: Wikepedia).