Selecting between causal and noncausal models with quantile autoregressions

3.1 QNCAR

A QNCAR(p) specification is introduced here as the noncausal counterpart of the QCAR(p) model by reversing time, explicitly as follows:

Analogously to the QCAR(p), the estimation of the QNCAR(p) goes through solving

with

where for the simplicity of notations, we use θ ˆ ( τ ) to denote the estimate in quantile noncausal autoregression. Drawing on the asymptotic derived by Koenker and Xiao (2006), we present the following theorem for QNCAR(p) based on three assumptions which are made to ensure covariance stationarity of the time series (by (A1) and (A2)) and the existence of quantile estimates (by (A3)).

The above result can be further simplified into Corollary 2 by adding the following assumption:

1. The coefficient matrix A _t in (12) is constant over time. (We denote A ≔ [ ϕ 1 ϕ 2 … ϕ p I p − 1 0 ( p − 1 ) × 1 ] for A _t under this assumption.)

As can be seen, QCAR(p) and QNCAR(p) generalize the classical purely causal and purely noncausal models respectively by allowing random coefficients on lag or lead regressors over time. Corollary 2 provides additional results when the same coefficients except the intercept are used to generate each quantile. However, the moment requirement in (A1) is very strict for heavy tailed time series. In order to study noncausality by QAR in heavy tailed distributions, we have to show its applicability when weakening the assumption (A1). This goal is achieved by Theorem 3 which presents the asymptotic behaviour of the QAR estimator for a classical purely noncausal model. Similarly, the asymptotic in a classical purely causal model follows right after reversing time.

1. { ε t } t = 1 n are i.i.d. innovation variables with regularly vary tails defined as

   (15) P ( | ε t | > x ) = x − α L ( x ) ,

   where L(x) is slowly varying at ∞and 0 < α < 2. There is a sequence {a _T } satisfying

   (16) T ⋅ P { ‖ ε t ‖ > a T   x } → x − α   f o r   a l l   x > 0.

   with b T = E [ ε t   I [ | ε t | ≤ a T ] ] = 0 . ^[4] The distribution function of ε _t , denoted as F(⋅), has continuous density f(⋅) with f(ε) > 0 on {ε:0 < F(ε) < 1} in probability one;

2. The roots of the polynomial ϕ(z)are greater than one, such that y _t can be written into

   (17) y t = ∑ j = 0 ∞ c j   ε t + j ,

   where ∑ j = 0 ∞ j   | c j | δ < ∞ for some δ < α, δ ≤ 1.

Then

where ϕ τ ≔ [ F − 1 ( τ ) a T , ϕ 1 , … , ϕ p ] , Ω: = [ω _ik ] being a _p×p matrix with entry ω i k ≔ ∑ j = 0 ∞ c j   c j + | k − i | at the ith row and the kth column, {S _α (s)}being a process of stable distributions with index α which are independent of Brownian motions {W(s)}. In this theorem the intercept regressor in QNCAR(p) is changed to a _T such that x′t: = [a _T , y _t , y _t+1, … , y _t+p−1].

Proof. See the appendix

Heuristically, next we restrict our focus on the classical models and explore consequences of causality misspecification in quantile regressions.

3.2 Causal and noncausal models with Gaussian i.i.d. disturbances

Suppose a causal AR(1) process { y t } t = 1 T , y _t  = α + βy _t−1 + ε _t , with for instance [α, β] = [1, 0.5], i.i.d. standard normal {ε _t } and T = 200. Figure 2 displays a simulated sample following this data generating process (DGP).

Figure 2:

Simulation of a one-regime process with N(0, 1) innovations, T = 200.

The information displayed in Figure 3 is the SRAR(τ) of each candidate model along quantiles, indicating their goodness of fit. The two SRAR curves almost overlap at every quantile, which implies no discrimination between QCAR and QNCAR in Gaussian innovations, in line with results in the OLS case. The Gaussian distribution is indeed time reversible, weak and strict stationary. Its first two moments characterize the whole distribution and consequently every quantile. Note that we obtain similar results for a stationary noncausal AR(p) process with i.i.d. Gaussian {ε _t }. The results are not reported to save space.

Figure 3:

SRAR plot under an AR(1) with N(0, 1) innovations, T = 200.

3.3 Causal and noncausal models with Student’s t distributed innovations

Things become different if we depart from Gaussianity. Suppose now a causal AR(1) process y _t  = α + βy _t−1 + ε _t with again [α, β] = [1, 0.5] but where {ε _t } are i.i.d. Student’s t−distributed with 2 degrees of freedom (hereafter using shorthand notation: t(2)). Figure 4 depicts a simulation in this AR(1) with T = 200. Applying QCAR and QNCAR respectively on this series results in the SRAR curves displayed in Figure 5. The distance between the two curves is obvious compared to the Gaussian case, favouring the causal specification at almost all quantiles. Figure 6 is the SRAR plot of a purely noncausal process with i.i.d. Cauchy innovations. The noncausal specification is preferred in the SRAR comparison.

Figure 4:

Simulation of an AR(1) with t(2) innovations, T = 200.

Figure 5:

SRAR plot under an AR(1) with t(2), T = 200.

Figure 6:

SRAR plot under a noncausal model with Cauchy innovations, T = 200.

Figure 7:

Simulation of a noncausal model with Cauchy innovations, T = 200.

It seems now that applying the SRAR comparison at one quantile, such as the median, is sufficient for model identification, but it is not true in general. In Section 4, we will spot an identification issue in the SRAR plots, the true model even having higher SRAR values at certain quantiles than the misspecified model.

So far we have applied QCAR and its counterpart QNCAR on the classical purely causal or noncausal models with symmetrically i.i.d. innovations. Within this restricted scope, the conditional mean models of those data generating processes only differ from their conditional quantile models in the intercept term. And we see that model selection by the SRAR comparison gives uniform decisions along quantiles. However, such a model selection is not always that clear in practise. For example, in the empirical study later, we will encounter an identification issue which can be seen by checking the SRAR plots. In the next section, we will present this identification issue with some possible reasons, and propose a robust model selection criterion called the aggregate SRAR to cope with this issue.

4.1 Identification issue spotted from the SRAR plots

First let us see some possible model settings causing the identification problem in SRAR plots. The first case is linked to the existence of multi-regimes in coefficients.

Suppose a regime-shift model is specified as follows:

where {ε _t } is an i.i.d. innovation process with cumulative probability function F(⋅), and β _t is defined as follows:

with τ * ∈ ( 0 , 1 ) and β ₁ < β ₂. In essence, the regime shift of β _t depends on the quantile occurrence of ε _t which is indexed by τ _t : = F(ε _t ) with {τ _t } being i.i.d. in the standard uniform distribution.

If {y _t } can be negative, then there is a problem in using QNCAR to recover the coefficients in the underlying model (19) because the τ-th regime is not necessary to produce the τ-th conditional quantile of y _t . So the comonotonicity condition of the linear quantile regressions (19) is not satisfied. However, by restricting to the non-negative region of the covariate y _t+1 (also see Fan and Fan 2010) we force the regression model to satisfy the comonotonicity requirement without losing its association with {τ _t }. The obtained estimator is also consistent to the true coefficients in (19).^[5] QCAR (or QNCAR) with such a restriction, hereafter denoted as RQCAR (or RQNCAR) shorthand for restricted quantile causal autoregression (or restricted quantile noncausal autoregression), is formulated as follows:

where I is the set restricting the quantile regression on a particular information set. The restriction is usually imposed in order for quantile regressions to meet the comonotonicity condition. To study the regime-shift model (19), we restrict the QNCAR on non-negative covariates, i.e. I = t : x t ≥ 0 .

Figure 8 shows four SRAR curves estimated from QCAR, QNCAR, RQCAR and RQNCAR. We consider a time series { y t } t = 1 600 simulated from the model (19) with τ ^* = 0.7, β ₁ = 0.2, β ₂ = 0.8 and i.i.d. innovation process in t(3), i.e. F ⁻¹(⋅) = F _t(3) ⁻¹(⋅).

Figure 8:

Identification problem spotted in the SRAR plot for restricted quantile autoregressions.

Figure 8 illustrates such an identification issue in which the SRAR curve from a true model is not always lower than one from misspecification. Applying restriction helps to enlarge the SRAR difference between a true model and a misspecified time direction.

The second case we investigate is the presence of skewed distributed innovations.

Let us consider a time series {y _t } following a purely noncausal AR(1): y _t  = 0.8y _t+1 + ε _t with {ε _t } i.i.d. in a demeaned skewed t distribution with skewing parameter γ = 2 and v = 3 degrees of freedom (hereafter t(v, γ) is the shorthand notation for a skewed t-distribution). The probability density function of t(v, γ) (see Francq and Zakoïan 2007) is defined as

where f _t (⋅) is the probability density function of the symmetric t(v) distribution. Figure 9 shows four SRAR curves obtained from the estimation of the QCAR, the QNCAR, the RQCAR and the RQNCAR respectively. The curves from the QNCAR and the RQNCAR almost overlap each other, which confirms our understanding that the monotonicity requirement is met in the true model. The estimations and the corresponding SRAR curves should be the same unless many observations are omitted by the restriction. On the other hand, the SRAR curve gets enlarged from the QCAR to the RQCAR, which is very reasonable as the feasible set in the QCAR is larger and the misspecification is not ensured to satisfy the monotonicity requirement. Again we see this identification problem from the SRAR plot. Remarkably, the SRAR curve from a true model can be higher at certain quantiles than the one from a misspecified model. Consequently the SRAR comparison relying only on particular quantiles, such as the least absolute deviation (LAD) method for the median only is not robust in general. Therefore, we propose a new model selection criterion in next subsection by including the information over all quantiles.

Figure 9:

Identification issue spotted from the SRAR plot for a skewed distribution.

4.2 The aggregate SRAR criterion

Based on the same number of explanatory variables in QCAR and QNCAR with a fixed sample size in quantile regressions, the best model is supposed to exhibit the highest goodness of fit among candidate models. Similarly to the R-squared criterion in the OLS, when turning to quantile regressions, we are led to use the SRAR criterion for model selection. The aggregate SRAR is regarded as an overall performance of a candidate model over all quantiles such as:

There are many ways to calculate this integral. One way is to approximate the integral by the trapezoidal rule. Another way is to sum up SRARs over a fine enough quantile grid with equal weights. In other words, this aggregation is regarded as an average of performances (SRAR(τ), τ∈(0, 1)) of a candidate model. In practise, there is almost no difference in model selection between the two aggregation methods.

As equal weights are used on all quantiles in the aggregate SRAR above, people may argue to use a different weighting scheme. The weighting scheme indeed can be different as when weight being one for one quantile and zero for others is to select a conditional quantile model. We agree that the weighting scheme can be customized in justice of users’ purpose. The equal-weight scheme proposed here is inspired to calculate the area under the SRAR curve over quantiles when we check the SRAR plot. Intuitively, such areas are directly linked to model performance when we concerns the whole dynamics of the underlying process. And we compare models by viewing the gap between their SRAR curves, which is the difference between the areas under their SRAR curves. This leads us to the aggregate SRAR measure.

Performances of the SRAR model selection criteria in Monte Carlo simulations are reported in Table 1. It shows the average frequencies with which we find the correct model based on the SRAR criterion per quantile and the aggregate SRAR criterion. The sample size T is 200 and each reported number is based on 2000 Monte Carlo simulations. Columns of Table 1 refer to as a particular distribution previously illustrated in this paper. As observed, the aggregate SRAR criterion performs very well even in situations with the identification issue. The Gaussian distribution being weakly and strictly stationary we cannot obviously discriminate between causal and noncausal specifications leading to an average frequency of around 50% to detect the correct model.

4.3 Shape of SRAR curves

By observing SRAR plots, we see that SRAR curves vary when the underling distribution varies. It is interesting to investigate the reasons. In this subsection, we will provide some insights on the slope and concavity of SRAR y t ( τ , θ ˆ ( τ ) ) curves under assumptions (A1), (A2), (A3) and (A4). Since ρ _τ (y _t − x _t ^’ θ ) is a continuous function in θ ∈ R ( p + 1 ) , by the continuous mapping theorem and τ , θ ˆ ( τ ) ) , we know that

We also know that

Therefore instead of directly deriving the shape of a SRAR y t ( τ , θ ˆ ( τ ) ) curve, we look at the properties of its intrinsic curve SRAR _εt (τ, F ⁻¹(τ)). We derive the first and second order derivatives of SRAR _εt (τ, F ⁻¹(τ)) with respect to τ in order to determine the shape of SRAR y t ( τ , θ ˆ ( τ ) ) .

6.1 The model specification

The motivation of our empirical analysis comes from the rational expectation (RE) hyperinflation model originally proposed by Cagan (1956) and investigated by several authors (see e.g. Adam and Szafarz 1992; Broze and Szafarz 1985). We follow Broze and Szafarz (1985) notations with

In (33), m _t ^d and p _t respectively denote the logarithms of money demand and price, x _t is the disturbance term summarizing the impact of exogenous factors. E(p _t+1|I _t ) is the rational expectation, when it is equal to conditional expectation, of p _t+1 at time t based on the information set I _t . Assuming that the money supply m _t ^s  = z _t is exogenous, the equilibrium m _t ^d  = m _t ^s provides the following equation for prices

Broze and Szafarz (1985) show that a forward-looking recursive solution of this model exists when x _t is stationary and | ϕ | < 1. The deviation from that solution is called the bubble B _t with p t = ∑ i = 0 ∞ ϕ i E ( u t + i | I t ) ] + B t . Finding conditions under which this process has rational expectation equilibria (forward and or backward looking) is out of the scope of our paper. We only use this framework to illustrate the interest of economists for models with leads components. Under a perfect foresight scheme E(p _t+1|I _t ) = p _t+1 we obtain the purely noncausal model

with ε ˜ t = u t . In the more general setting, for instance when E(p _t+1|I _t ) = p _t+1 + v _t with v _t a martingale difference, the new disturbance term is ε ˜ t = v t + u t . Empirically, a specification with one lead only might be too restrictive to capture the underlying dynamics of the observed variables. We consequently depart from the theoretical model proposed above and we consider empirical specifications with more leads or lags. Lanne and Luoto (2013, 2017) and Hecq, Lieb, and Telg (2017a, 2017b) in the context of the new hybrid Keynesian Phillips curve assume for instance that ε ˜ t is a MAR(r−1, s−1) process such as

where ε _t is iid and c an intercept term. Inserting (35) in (34) we observe that if ε ˜ t is a purely noncausal model (i.e. a MAR(0, s−1) with ρ(L) = 1) we obtain a noncausal MAR(0, s) motion for prices

We would obtain a mixed causal and noncausal model if ρ(L) ≠ 1. Our guess that the same specification might in some circumstances empirically (although not mathematically as the lag polynomial does not annihilate the lead polynomial) gives rise to a purely causal model in small samples when the autoregressive part dominates the lead component.

The above illustration presents a context of pure causal and noncausal models so that we can apply our approach to give an empirical analysis. It would be interesting to extend our modelling to investigate theoretical models with both forward and backward behaviours such as backward- and forward-looking Taylor rule for instance. To do that however we have to introduce additional regressors and extend the approach of Hecq, Issler, and Telg (2020) to quantile regressions, which can be further investigated by future research and is out of the scope of this paper.

6.2 The data and unit root testing

We consider seasonally unadjusted quarterly Consumer Price Index (CPI) series for four Latin American countries: Brazil, Mexico, Costa Rica and Chile. Monthly raw price series are downloaded at the OECD database for the largest span available (in September 2018). Despite the fact that quarterly data are directly available at OECD, we do not consider those series as they are computed from the unweighted average over three months of the corresponding quarters. Hence, these data are constructed using a linear filter, leading to undesirable properties for the detection of mixed causal and noncausal models (see Hecq, Telg, and Lieb 2017a, b on this specific issue). As a consequence, we use quarterly data computed by point-in-time sampling from monthly variables. The first observation is 1969Q1 for Mexico, 1970Q1 for Chile, 1976Q1 for Costa Rica and 1979Q4 for Brazil. Our last observation is 2018Q2 for every series. We do not use monthly data in this paper as monthly inflation series required a very large number of lags to capture their dynamic feature. Moreover, the detection of seasonal unit roots in the level of monthly price series was quite difficult.

Applying seasonal unit root tests (here HEGY tests, see Hylleberg et al. 1990) with a constant, a linear trend and deterministic seasonal dummies, we reject (see Table 2 in which a * denotes a rejection of the null unit root hypothesis at a specific frequency corresponding to 5% significance level) the null of seasonal unit roots in each series whereas we do not reject the null of a unit root at the zero frequency. The implementation of the unit root tests here is concerned with conditional mean models of the raw data to ensure that we process the data and use its weakly stationary time series in quantile regressions for analysis. The unit root testing can also been done per quantile (Koenker and Xiao 2004) to relate short-term explosiveness of time series to unit-root quantile models, which is an interesting perspective to treat explosive time series and alternative to causal and noncausal modelling. We do not go deeper in the unit-root direction for this paper but with its outlook on future research.

The number of lags of the dependent variable used to whiten for the presence of autocorrelation is chosen by AIC. From these results we compute quarterly inflation rates for the four countries in annualized rate, i.e. Δ ln P _t ⁱ  × 400.Next we carry out a regression of Δ ln P _t ⁱ  × 400 on seasonal dummies to capture the potential presence of deterministic seasonality. The null of no deterministic seasonality is not rejected for the four series. Figure 13 displays quarterly inflation rates and it illustrates the huge inflation episodes that the countries had faced. Among the four inflation rates, Brazil and Mexico show the typical pattern closer to the intuitive notion of what a speculative bubble is, namely a rapid increase of the series until a maximum value is reached before the bubble bursts.

Figure 13:

Quarterly inflation rate series plot for 4 Latin American countries.

6.3 Empirical findings and identification of noncausal models

Table 3 reports for each quarterly inflation rates the autoregressive model obtained using the Hannan–Quinn information criterion. Given our results on the binding function (see also Gouriéroux and Jasiak 2017) it is safer to determine the pseudo-true autoregressive lag length using such an OLS approach than using quantile regressions or using maximum likelihood method. Indeed there is the risk that a regression in direct time from a noncausal DGP provides an underestimation of the lag order for some distributions (e.g. the Cauchy) and some values of the parameters.

Estimating autoregressive univariate models gives the lag length range from p = 1 for Brazil to p = 7 for the Chilean inflation rate. The p ₋ values of the Breush-Pagan LM test (see column labelled LM[1−2]) for the null of no-autocorrelation after having included those lags show that we do not reject the null in every four cases. On the other hand, we reject the null of normality (Jarque–Bera test) in the disturbances of each series. We should consequently be able to identify causal from noncausal models. From columns skew. and kurt. it emerges that the residuals are skewed to the left for Brazil and Mexico and skewed to the right for Chile and Costa Rica. Heavy tails are present in each series. At a 5% significance level we reject the null of no ARCH (see column ARCH[1−2]) for Brazil and Mexico. Gouriéroux and Zakoian (2017) have derived the closed form conditional moments of a misspecified causal model obtained from a purely noncausal process with alpha stable disturbances. They show that the conditional mean (in direct time) is a random walk with a time varying conditional variance in the Cauchy case. This result would maybe favour the presence of a purely noncausal specification for Brazil and Mexico as the null of no ARCH is rejected. But this assertion must be carefully evaluated and tested, for instance using our comparison of quantile autoregressions in direct and reverse time. The results by the Q(N)CAR are reported in Table 4, and the RQ(N)CAR produces the same results. Each cell of Table 4 provides the selection frequency of MAR(0, p) or MAR(p, 0) identified by the SRAR at quantiles 0.1, 0.3, 0.5, 0.7, 0.9 as well as the aggregated SRAR. Figure 14 displays the SRAR curves from 0.05th-quantile to 0.95th-quantile by the Q(N)CAR for the four economies respectively, similarly to the ones by the RQ(N)CAR with restriction on non-negative regressors. As observed, the identification problem is raised in the SRAR plots. Especially in the SRAR plot for Brazil, it is hard to trust a model from evidence at single quantiles. However, the aggregate SRAR criterion comes to help in this situation from an overall perspective. We conclude that Brazil, Mexico and Costa Rica are better characterized as being purely noncausal while Chile being purely causal according to the aggregate SRAR criterion.

Figure 14:

SRAR plots of the inflation rates of four Latin American countries respectively.