13.5.1 Estimation

13.5.1 Estimation

Following Koenker and Bassett (1978), the parameters of quantile models can be estimated by replacing the conventional quadratic loss function underlying most empirical work on predictability with the so-called “tick” loss function,

$$L_{\alpha }(e_{t+1}(\beta ))=(\alpha -1(e_{t+1}(\beta )<0))e_{t+1}(\beta ),\text{\hspace{1em}(13.47)}$$

where $e_{t+1}(\beta )=y_{t+1}-q_{\alpha }(y_{t+1}|Z_t,\beta )$ is the forecast error and $1(\cdot )$ is the indicator function. Under this objective function, the optimal forecast is the conditional quantile. To see this, note that the first-order condition associated with minimizing the expected value of (13.47) with respect to the forecast, $q_{\alpha }(y_{t+1}|Z_t,\beta )$ , is the $\alpha \cdot$ quantile of the return distribution,

$$q_{\alpha }(y_{t+1}|Z_t,\beta )=P_{t+1|t}^{-1}(\alpha ),\text{\hspace{1em}(13.48)}$$

where $P_{t+1|t}$ is the conditional distribution function for $e_{t+1}$ . In fact, for the forecast to be represented by the conditional α-quantile of $y,$ it is necessary for the forecaster’s loss to take the slightly more general form,

$$L(f,y)=[\alpha -1(G(y)\le G(f))]\left(G(y)-G(f)\right),$$

where $G$ is a strictly increasing function; see the discussion in Komunjer (2013). Under the loss function in (13.47), estimates can be obtained by solving a linear programming problem.

Conversely, specifications such as (13.46) are highly nonlinear functions in the parameters, θ, and so do not lend themselves to be solved by means of linear programming methods. Instead Markov chain Monte Carlo or minimax methods can be used; see Komunjer (2013) for further discussion.

Empirical evidence on the performance of various procedures for constructing quantile forecasts is provided in a number of studies. Kuester, Mittnik, and Paolella (2006) consider returns on the NASDAQ composite index over a 30-year period. They find that GARCH specifications generally underestimate the likelihood of extreme returns, particularly if the innovations are assumed to be Gaussian. Assuming instead that the innovations follow a skewed-t distribution leads to somewhat better results. Bao, Lee, and Saltoglu (2006) report shortcomings for CaViaR models fitted to five Asian stock markets during the Asian crisis in 1997–1998, although the models perform substantially better prior to the crisis.