Dynamic quantile causal inference and forecasting
- Autores: Josef Ruzicka
- Directores de la Tesis: Jesús Gonzalo Muñoz (dir. tes.)
- Lectura: En la Universidad Carlos III de Madrid ( España ) en 2021
- Idioma: español
- Tribunal Calificador de la Tesis: José Olmo Badenas (presid.), Victor Emilio Troster (secret.), Mario Alloza (voc.)
- Programa de doctorado: Programa de Doctorado en Economía por la Universidad Carlos III de Madrid
- Materias:
- Enlaces
- Tesis en acceso abierto en: e-Archivo
- Resumen
The first two chapters of thesis are devoted to dynamic quantile causal inference via impulse response functions. The first chapter uses an indirect, simulation-based approach, while the second chapter is based on direct estimation of impulse response functions. The third chapter is devoted to forecasting. Specifically, it modifies an estimator proposed in the second chapter and applies it in an environment with many potential predictors.
The first chapter is called “Quantile Structural Vector Autoregression”. Quantile regression of Koenker and Bassett (1978) is used in many disciplines because it is more robust and provides more information than the traditional least squares. However, applications of quantile methods in empirical macroeconomics are still scarce. Macroeconometrics involves dynamic multivariate systems and sophisticated identification strategies, which impede direct applications of existing quantile regression methods. We introduce a methodology that incorporates quantile information into the structural vector autoregressive model. Structural vector autoregression (SVAR) of Sims (1980) is the workhorse of empirical macroeconomics. It describes the conditional mean and it is driven by uncorrelated structural disturbances (shocks), which equal the difference between each observation and the conditional mean. They are often analyzed by impulse response functions, which describe the effect of an external impulse (shock) on the conditional mean of the response variables. Koenker and Xiao (2006) study quantile regression in univariate time series. Their model has been applied to inflation by Wolters and Tillmann (2015) and GDP growth by Liu (2019).
Several authors have recently applied quantile regression to vector autoregression. Montes-Rojas (2019a) uses a reduced form quantile vector autoregressive model to construct quantile impulse response functions for fixed sample paths. Chavleishvili and Manganelli (2017, 2019) study structural vector autoregressive models identified by recursive short-run restrictions and employ quantile impulse response functions based on fixed sample paths as in Montes-Rojas (2019a). Montes-Rojas (2019b) extends his previous work by simulating distribution of impulse response functions in a model identified by recursive short-run restrictions.
We extend the existing literature in two key respects. First, we design a coherent framework that encompasses popular SVAR identification strategies. In particular, we consider actual structural identification and not just recursive short-run restrictions. We introduce non-recursive short-run restrictions, long-run restrictions, external instruments, and their combinations. Second, we show how to make inference about quantile treatment effects of structural macroeconomic shocks. In particular, we establish weak convergence of the estimators and weighted bootstrap consistency. Doing so, we also apply the instrumental variable quantile regression (Chernozhukov and Hansen, 2006) in a time series setting.
We propose quantile structural vector autoregression (QSVAR) as a multivariate counterpart of quantile autoregression of Koenker and Xiao (2006). QSVAR is a system of structural quantile functions in the sense of Chernozhukov and Hansen (2008), differing from the usual conditional quantile functions. It is precisely this novel formulation which allows a range of identification schemes. The model describes the conditional distribution and is driven by independent structural disturbances which indicate quantiles within the distribution.
QSVAR models, just like SVAR, have many coefficients that interact in a complicated way. In order to trace the effects of structural shocks, we introduce a notion of quantile impulse responses. The quantile impulse response measures the effect of an external impulse on the conditional distribution of the response variables. It captures the dynamic causal effects of shocks on different parts of the distribution, on the interquartile range or on volatility, which the standard approach cannot do.
The SVAR arises from the Wold decomposition, which is not helpful in a quantile regression setting. However, the QSVAR model is not a mere combination of quantile regression and SVAR with no theoretical justification. We establish a structural quantile decomposition, which states that every multivariate stochastic process is given by a function of its past and current values and standard uniform disturbances. These disturbances are independent both over time and across components, allowing a structural interpretation. A linear approximation of this representation gives rise to the QSVAR process.
A simulation study shows the point estimators and confidence intervals perform well in small samples, typical in macroeconomic applications. We also find that when the process generates many outliers (heavy-tailed distributions), then the median treatment effect from QSVAR is more accurate than the average treatment effect from SVAR.
Our empirical applications reveal causal effects that cannot be obtained by standard SVAR. First, we find that the oil price shock has a negative and statistically significant effect on GDP growth only in the left tail of GDP growth distribution. This means a spike in oil price can cause a recession, but there is no evidence that a drop in oil price could cause an expansion. Second, the government spending multiplier is asymmetric across the GDP growth distribution: it is larger in its right part. Thus, positive fiscal shocks may cause economic growth, but there is no evidence negative fiscal shocks may cause economic decline. Third, the tax multiplier is positive and statistically significant across the entire GDP growth distribution. This contrasts with the SVAR, where the tax multiplier is not statistically significant, as the estimate is less accurate. Fourth, we find that the real activity shock reduces dispersion of stock market returns and that the real activity shock is more important for explaining stock market falls than rallies.
The first chapter is organized as follows. Subsection 1.2 lays out the foundations: SVAR and quantile autoregression, as well as the structural quantile decomposition, which gives rise to the QSVAR model. Subsection 1.3 formally defines the QSVAR and the quantile impulse responses and explores their properties. Subsection 1.4 is concerned with identification, subsection 1.5 with estimation and inference. Finite sample properties of the point estimates and confidence bands are documented in subsection 1.6, which also shows specific examples of the QSVAR process. Four empirical applications are given in subsection 1.7. Subsection 1.8 concludes. The appendix provides proofs, additional simulation results and complete figures to empirical applications.
The second chapter, “Quantile Local Projections: Identification, Smooth Estimation, and Inference”, has the same objective as the first chapter: to capture quantile treatment effects of macroeconomic shocks while taking into account the essential aspects of structural identification and inference. However, it relies on a different methodological approach. Impulse response functions in the second chapter are not constructed by simulation, but they are being estimated directly.
Quantile regression (QR), introduced by Koenker and Bassett (1978), is popular in many disciplines because it captures more information than the least squares regression, while also being more robust. In empirical macroeconomics, these two features are particularly important: The samples tend to be small (hence the need to use all available information), policy makers’ preferences over the outcomes are asymmetric (so some parts of the distribution are more important than the mean), and outliers abound. These considerations have led to a surge of QR applications in recent years.
One of the main endeavors in macroeconometrics is the measurement of dynamic causal effects of shocks – impulse response analysis. A simple and popular impulse response method are the local projections of Jordà (2005). Compared to the more traditional approach based on vector autoregression, the local projections offer various advantages. Local projection impulse responses are given by regression coefficients, leading to simple estimation and inference. They are also more robust to misspecification and can easily incorporate nonlinearities. However, standard local projections, which are based on a least squares regression, only capture the average effects of shocks. An increasingly popular way to capture the heterogeneity of impulse response functions are quantile local projections, which consist in estimating local projections by quantile regression.
Nevertheless, empirical researchers using quantile local projections face three important challenges in practice. First, it is not clear how to identify the dynamic causal effects of interest. Different authors use different specifications, for instance, some control for lagged values, others control for current values. Second, the impulse responses often wiggle a lot, which is at odds with macroeconomic theory and it also hampers their interpretation. Third, it is unclear how to make valid inference in the quantile local projections context.
The main contribution of the second chapter is a statistical framework for quantile local projections, which encompasses identification, smooth estimation, and inference. We show how to identify dynamic quantile causal effects by popular schemes: short-run restrictions, external instruments, and long-run restrictions. We also introduce two novel estimators, called Smooth Quantile Projections (SQP) and Smooth Quantile Projections with Instruments (SQPI). While the SQP estimator is identified by recursive short-run restrictions similar to Sims (1980), the SQPI is identified by external instruments or long-run restrictions. In addition, the SQP estimator delivers multi-horizon conditional quantile forecasts, which are smooth over forecast horizons. The usefulness of the framework is demonstrated in the setting of Adrian et al. (2019), whereby we reach a different conclusion – we show that financial conditions affect the entire distribution of future GDP growth and not just its lower part. The proposed estimators rely on the following estimation principle: The conditional quantiles of the response variable should be smooth functions of the forecast horizon. We impose this smoothness by total variation roughness penalties like Koenker et al. (1994).
However, while they smooth across the dimension of the explanatory variable, we are smoothing across the horizon of the response variable. This difference has implications for asymptotics as our estimator retains the parametric convergence rate, in contrast to the one of Koenker et al. (1994), which has a slower rate of convergence (Portnoy, 1997). We provide closed-form confidence intervals for our estimators as well as weighted bootstrap procedures. The SQPI confidence intervals are valid even when instrument’s relevance condition fails, owing to the dual inference of Chernozhukov and Hansen (2008) that we adopt. This is convenient because the instruments used in local projections may give rise to weak instrument concerns (Ganics et al., 2021). Our estimators are computationally convenient: While the SQP estimator solves a linear program, the SQPI estimator solves a collection of inverse quantile regressions with a single endogenous variable in combination with a shortest path algorithm.
The penalties ensure that the estimated coefficients, as functions of horizon, are piecewise polynomial functions of a given degree, whose roughness is controlled by the penalty. When the penalties are set to zero, then the SQP estimator reduces to local projections estimated by QR, and the SQPI estimator coincides with local projections estimated by instrumental variable quantile regression (Chernozhukov and Hansen, 2005). In the least squares setting, Barnichon and Brownlees (2019) introduce smooth local projections, which merge two smoothing approaches: a transformation of explanatory variables through a two-sided moving average whose coefficients are determined by a B-spline, and a ridge regression shrinkage towards a polynomial. Our estimators can be seen as the (asymmetric) l1 counterpart of their second smoothing approach; we do not transform the regressors.
Many recent studies employ QR in local projections. The earliest reference seems to be Distante et al. (2013), who document asymmetric effects of technology shocks. Another example is Linnemann and Winkler (2016), who focus on fiscal policy. Adrian et al. (2019) highlight the importance of financial conditions for future declines in economic activity using a two-step procedure. In the first step, they estimate local projections by QR. In the second step, they match selected quantiles of the estimated conditional quantile functions with those of skewed t-distribution. Our approach is different: While they smooth across the dimension of the response variable by imposing a distributional assumption, we smooth across horizons nonparametrically. A different methodology, particularly suitable for financial applications involving Value-at-Risk, is developed by Han et al. (2019), who introduce local projections in the MVMQ-CAViaR model of White et al. (2015).
A simulation study shows that the SQP estimator with roughness penalty determined by an information criterion produces more accurate impulse responses at given quantiles than the plain quantile regression local projections. In small samples, the weighted bootstrap confidence intervals are more reliable than the closed-form ones.
The empirical applications demonstrate that the absence of smoothing leads to very volatile impulse response functions, causing type I errors. The smoothing effectively eliminates this volatility. In contrast to Adrian et al. (2019), the SQP empirical application shows that financial conditions matter for the entire distribution of GDP growth and not just for its lower quantiles. In addition, our setting is more flexible as we do not impose any specific distribution. Our findings in the SQPI empirical application contrast with Gertler and Karadi (2015), but they are in line with Stock and Watson (2018). However, out inference is robust to a weak instrument, it is more robust to outliers, and it does not require Gaussianity of structural shocks.
The second chapter is structured as follows. Section 2.2 covers the statistical framework and impulse response notion. Sections 2.3 and 2.4 introduce the SQP and the SQPI estimators, respectively, and address their identification, estimation, inference, and roughness penalty selection. Section 2.5 is a simulation study. Section 2.6 provides two empirical applications. Section 2.7 concludes. Proofs and some computational aspects are in the appendix 2.8.
The third chapter, “Smooth Quantile Projections in a Data-rich Environment”, is devoted to distribution forecasts. Distribution forecasts play an important role in a range of situations, including macroeconomic projections (fan charts), empirical asset pricing (asset return predictability), risk management, stress testing and climate change. In contrast to the simple mean forecasts, the distribution forecasts are much more informative – they can be used to make density forecasts, but also to forecast different moments. A popular way of constructing distribution forecasts is via quantile regression (QR) of Koenker and Bassett (1978), which has several advantages. As opposed to parametric approaches, hinging on a particular distribution (such as normal), QR is a flexible, semiparametric method, which does not require arbitrary distribution choices. In addition, QR is very convenient computationally and it is robust to outliers.
There are two basic approaches for constructing QR forecasts. The first is based on simulation (Koenker and Xiao, 2006; Koenker et al., 2018), while the second consists in direct forecasts, that is, estimating different models for different forecast horizons. The direct forecasting has become the dominant approach, since it is easy to implement and in contrast to the first approach, it is not computationally intensive. However, the direct forecast approach has one major drawback – the resulting forecasts often vary wildly between neighboring forecast horizons, which makes them difficult to interpret, but it also suggests that they are not estimated accurately.
In a related context of impulse response functions, Ruzicka (2021) introduces the Smooth Quantile Projections (SQP) estimator. The SQP estimator is based on the following estimation principle: The conditional quantile functions should be smooth functions of the forecast horizon. The estimation is carried out over various forecast horizons jointly, subject to roughness penalties, with shrinkage towards polynomials of a given degree. Although Ruzicka (2021) focuses on impulse response functions, he conjectures the forecasts based on the SQP may offer the following three advantages relative to plain QR: They are more accurate, easier to interpret, and less prone to the quantile crossing problem.
The third chapter has two main objectives. First, to assess the advantages of using the SQP estimator for forecasting through an out-of-sample evaluation exercise. Second, to introduce a new estimator, SQP with lasso, which modifies the original SQP estimator by introducing lasso penalties (Tibshirani, 1996). The novel estimator is computationally tractable and it is particularly suitable for environments with a large number of variables, increasingly relevant in practice. We propose information criteria for optimal penalty selection, but also implement cross-validation.
We carry out an out-of-sample evaluation of the new estimator and compare its performance with simple QR with lasso, employed e.g. by Manzan (2015), as well as with quantile autoregression. We show that the new estimator outperforms the other two methods. A more extensive exercise that would compare other estimation approaches, including factors, is left for future research.
Our work falls into the broader literature on quantile forecasting with large data sets. For instance, Lima and Meng (2017) show that estimating quantile regressions with lasso at different quantiles and subsequently averaging them improves equity premium forecasts. Carriero et al. (2020) compare various methods, including quantile regression with lasso, for nowcasting economic activity. Chen et al. (2021) introduce quantile factors and illustrate their usefulness in density forecasts of GDP growth and inflation.
