5.1 Linear dynamic quantile regression models

We are now in a position to introduce a quite general linear dynamic quantile regression model. Assume $\big((Y_t,Z_t)\colon\,t\in\mathbb{Z}\big)$ is a stochastic process where, for each $t,$ $Y_t$ is a scalar random variable and $Z_t$ is a ${\mathrm{D}_Z}$-dimensional random vector, and suppose that, for $\tau\in(0,1)$ and $t\in\mathbb{Z},$ the following time homogeneous quantile regression equation is satisfied, \[\begin{equation} Q_{Y_{t}}(\tau|\mathfrak{F}_{t-1}) = \alpha_0(\tau) + \sum_{j=1}^{\mathrm{lag}_Y}\alpha_{j}(\tau) g_j(Y_{t-j}) + \sum_{\ell=1}^{\mathrm{lag}_Z}Z_{t-\ell}^\prime\theta_\ell(\tau) \tag{5.2} \end{equation}\] for some (unknown) functions $\alpha_j\colon(0,1)\to\mathbb{R},$ $j\in\{0,\dots,{\mathrm{lag}_Y}\}$ and $\theta_\ell\colon(0,1)\to\mathbb{R}^{\mathrm{D}_Z},$ $\ell\in\{1,\dots,{\mathrm{lag}_Z}\},$ and some (typically known) functions $g_j,$ $j\in\{1,\dots,{\mathrm{lag}_Y}\}.$ In view of the discussion in the preceding sections, we can assume for convenience that $\operatorname{support}(Z_t)\subseteq[0,1]^{\mathrm{D}_Z}$ and that the functions $g_j$ map the real line into the unit interval (so $\operatorname{support}(g_j(V))\subseteq[0,1]$ for any random variable $V$ and all $j$), in order to ensure the non-negativity and boundedness which in turn allow for greater flexibility on the functional forms of the $\alpha$’s and $\theta$’s in (5.2).¹⁹

Exercise 5.3 Show that, for any random variable $Y$, if $G\colon\mathbb{R}\to\mathbb{R}$ is an increasing function, then $Q_Y = G^{-1}\circ Q_{G(Y)}.$ Conclude that substituting $Q_{Y_t}(\tau|\mathfrak{F}_{t-1})$ by $Q_{G(Y_t)}(\tau|\mathfrak{F}_{t-1})$ in (5.2), with $G$ as above, does not really add generality to the model. $\blacksquare$

Now, if for a fixed $t$ we write $Y := Y_t,$ \[ X := \begin{pmatrix} 1 & g_1(Y_{t-1}) & \cdots & g_{\mathrm{lag}_Y}(Y_{t-{\mathrm{lag}_Y}})&Z_{t-1}^\prime & \cdots & Z_{t-{\mathrm{lag}_Z}}^\prime\end{pmatrix}^\prime \] and $\beta := (\alpha_0\quad \alpha_1\,\cdots\, \alpha_{\mathrm{lag}_Y}\quad \theta_1^\prime\,\cdots\,\theta_{\mathrm{lag}_Z}^\prime),$ then — voilà — we are back to the (hopefully familiar by now) standard, static linear quantile regression model of the previous sections. Indeed, with this notation we have \[ Q_{Y_t}(\tau|\mathfrak{F}_{t-1}) = Q_{Y|X}(\tau|X) = X^\prime\beta(\tau),\quad\tau\in(0,1). \]

Exercise 5.4 Let $(W_t)_{t\ge0}$ be a Markov chain with state space $S=\{0,1\}$, initial distribution $\lambda_W$ and transition matrix \[ P = \begin{pmatrix} p_{00} & p_{01}\\ p_{10} & p_{11} \end{pmatrix}. \]

Also, let $(Y_t)_{t\ge0}$ be a stochastic process with state space $S$, initial distribution $\lambda_Y$ and assume that, for $\tau\in(0,1)$ and $t\ge1$, the following holds, \[ Q_{Y_{t}}(\tau\,|\,\mathfrak{F}_{t-1}) = \alpha_0(\tau) + \alpha_1(\tau)Y_{t-1} + \theta_1(\tau)W_{t-1} + \theta_2(\tau)W_{t-1}Y_{t-1} \] where $\mathfrak{F}_{t} = \sigma\big((Y_s,W_s)\colon\,s\le t\big)$ and where, for $\tau\in(0,1)$, the coefficients $\alpha_1, \alpha_2, \theta_1$ and $\theta_2$ are defined by \[\begin{align*} \alpha_0(\tau) &= \mathbb{I}_{[1/4\le \tau<1]}\\ \alpha_1(\tau) &= \mathbb{I}_{[1/4\le\tau<1/2]}\\ \theta_1(\tau) &= \mathbb{I}_{[1/4\le\tau<1/2]}\\ \theta_2(\tau) &= \mathbb{I}_{[1/4\le\tau<1/2]} - \mathbb{I}_{[1/2\le\tau<3/4]}. \end{align*}\]

Show that, conditional on $\mathfrak{F}_{t}$, the random variable $Y_{t+1}$ follows a $\mathsf{Bernoulli}(V_t)$ distribution, where \[\begin{equation*} V_t = \begin{cases} 1/4 & \textsf{if $W_t=0$ and $Y_t=0$}\\ 1/2 & \textsf{if ($W_t=1$ and $Y_t=0$) or ($W_t=0$ and $Y_t=1$)}\\ 3/4 & \textsf{if $W_n=1$ and $Y_n=1$} \end{cases} \end{equation*}\]
Show that $(Y_t,W_t)_{t\ge0}$ is a Markov chain with state space $S\times S$ and find the respective transition matrix.

This exercise provides a very simple example of a stochastic process $\big((Y_t,Z_t)\colon t\ge0\big),$ where $Z_t=(W_t\quad W_tY_t)^\prime,$ satisfying the time homogeneous quantile regression equation (5.2). $\blacksquare$

When the functions $g_1,\dots,g_{\mathrm{lag}_Y}$ are all equal to the identity function, the model described in equation (5.2) is precisely the QADL²⁰ model of Galvao, Montes-Rojas, and Park (2013), up to the definition of $\mathfrak{F}_t$ and inclusion of a contemporaneous $Z_t$ (notice, however, that the “minimum lag of $Z$” is arbitrary: we can always define $\tilde{Z}_{t} = Z_{t-1}$ in which case equation (5.2) includes a contemporaneous $\tilde{Z}_t$). If additionally the $\theta$’s are all equal to the zero function, then (5.2) corresponds to the QAR²¹ model of Koenker and Xiao (2006). Some remarks: first, with this notation the QADL model as introduced by Galvao, Montes-Rojas, and Park (2013) would have $\sigma\big\{(Y_s,\tilde{Z}_s)\colon s\le t\big\}$ in place of $\mathfrak{F}_{t-1},$ which is likely a typo since $Y_t$ behaves like a constant when conditioned on itself. Second, it is also important to keep in mind that equation (5.2) does not fully specify the dynamic behavior of the multivariate time series $\big((Y_t,Z_t)\colon\,t\in\mathbb{Z}\big).$ Rather, it determines a class of models. This is a feature prevalent when dynamics are introduced via stochastic difference equations, for example those characterizing the class of $\mathsf{VAR}(1)$ models, in which (as mentioned earlier) a solution consists of a probability law of a stochastic process which obeys said equations. The solutions (which may or may not be stationary) will depend on the parameter matrix and on assumptions about the distribution of the innovation terms. Such a feature is particularly salient when it comes to the QADL equations, as there might be more than one probability law for $\big((Y_t,Z_t)\colon\, t\in\mathbb{Z}\big)$ according to which (5.2) holds, even for the same set of functional parameters (the $\alpha$‘s and $\theta$’s). The piece that is missing here (in order to fully specify the dynamics of the process, up to the functional parameters) are the conditional distributions $\mathbf{P}[Z_t\in \cdot\,|\,\mathfrak{F}_{t-1}, Y_t],\,t\in\mathbb{Z}$. Third, one can find in the literature several time series models for bounded random variables. In many cases, these fall into a fully parametric framework (for example the $\beta$ARMA model) — hence, if one is interested in estimating the conditional quantile functions, the “correct” procedure would be to estimate the model parameters via (quasi/pseudo)maximum likelihood,²² and then recover the desired conditional quantile functions by plugging in the estimated values into the (known, up to parameters) formula for the theoretical distribution. In general these parametric models will have a conditional quantile function which is non-linear in the parameters.

Example 5.1 Assume that $Y_t|\mathfrak{F}_{t-1}$ has a Kumaraswamy distribution with parameters $Y_{t-1}/\alpha$ and $Y_{t-1}/\beta$, where $\alpha,\beta>0$. The quantile function of a random variable $V$ having the Kumaraswamy distribution with parameters $a>0$ and $b>0$ is given by \[ Q_V(\tau)=(1-(1-\tau)^\frac{1}{b})^\frac{1}{a},\quad\tau\in(0,1), \] so our assumption is that \[ Q_{Y_t}(\tau|\mathfrak{F}_{t-1}) = (1-(1-\tau)^\frac{\alpha}{Y_{t-1}})^\frac{\beta}{Y_{t-1}},\quad\tau\in(0,1) \] This model is not nested in (5.2).

In order that everything is well posed, we also have to preclude linear dependence between the random variables appearing as predictors in the right-hand side of (5.2) (including the constant random variable). In fact, we need to require some additional technical measurability constraints as well.↩︎
For Quantile autoregressive distributed lag.↩︎
Quantile autoregressive, but you had already guessed this one.↩︎
As we will see later, estimation of the parameters in (5.2) is achieved by a different optimization problem.↩︎