class: center, middle, inverse, title-slide # Forecasting for Economics and Business ## Lecture 10: Multistep Forecasting Methods ### David Ubilava ### University of Sydney --- # Multistep Forecasting Methods Consider an AR(1): `\(y_t=\alpha+\beta y_{t-1}+\varepsilon_t.\)` From this, the one-step-ahead forecast is readily given by: `$$y_{t+1|t}=\alpha+\beta y_{t}.$$` We can obtain a two-step-ahead forecast using the *iterated* method: `$$y_{t+2|t}=\alpha+\beta y_{t+1|t}.$$` Alternatively, we can substitute the expression of `\(y_{t+1|t}\)` in here to obtain: `$$y_{t+2|t}=\alpha+\beta (\alpha+\beta y_{t}) = \alpha(1+\beta)+\beta^2y_t.$$` Thus, we obtain a two-step-ahead forecast using the *plug-in* method. --- # Multistep Forecasting Methods Similarly, we can substitute `\(y_{t-1}=\alpha+\beta y_{t-2}+\varepsilon_{t-1}\)` into the original equation to obtain: `$$y_t=\alpha(1+\beta)+\beta^2y_{t-2} + \varepsilon_t + \beta\varepsilon_{t-1} = \tilde{\alpha} + \tilde{\beta} y_{t-2} + u_t,$$` where `\(\tilde{\alpha}=\alpha(1+\beta)\)` and `\(\tilde{\beta}=\beta^2\)`, and `\(u_t=\varepsilon_t + \beta\varepsilon_{t-1}.\)` Thus, a way to obtain two-step-ahead forecast is if we regress `\(y_t\)` on `\(y_{t-2}\)`, and then use the parameter estimates to *directly* forecast `\(y_{t+2}\)`. This method is referred to as the *direct* method of forecasting. The approach can be extended to higher order autoregressive models, as well as to Multistep forecasts at any horizon. --- # Multistep Forecasting Methods To summarize, an `\(h\)`-step-ahead forecast from an AR(p) model, using: - the iterated method: `$$y_{t+h|t,i} = \alpha + \sum_{j=1}^{p}\beta_j y_{t+h-j|t,i},$$` where `\(y_{t+h-j|t,i}=y_{t+h-j}\)` when `\(h-j\le 0.\)` - the direct method is: `$$y_{t+h|t,d} = \tilde{\alpha} + \sum_{j=1}^{p}\tilde{\beta}_j y_{t+1-j}$$` - the plug-in method can take a rather cumbersome expression as the order of autoregression and the horizon length increase. --- # Direct vs Iterated Multistep Forecasts The relative performance of the two forecasts, `\(y_{i,t+h|t}\)` and `\(y_{d,t+h|t}\)`, in terms of bias and efficiency depends on the bias and efficiency of the estimators of each method. In the case of correctly specified models, both estimators are consistent, but the one-step model (which leads to the iterated method) is more efficient. Thus, in large samples, the iterated forecast can be expected to perform better than the direct forecast. In the case of mis-specified models, the ranking may very well change. --- # Direct vs Iterated Multistep Forecasts The residual of the regression for direct method, `\(u_t\)`, while uncorrelated with the right-hand-side variables, is serially correlated. For example, in the case of the direct two-step-ahead forecast from a re-specified AR(1) model: `$$u_t = \varepsilon_t+\beta\varepsilon_{t-1}.$$` It then follows that `\(Var(u_t) = Var(\varepsilon_t+\beta\varepsilon_{t-1}) = \sigma^2(1+\beta^2).\)` Note that this is also the expression of the two-step-ahead forecast error variance from an AR(1) model. Thus, in the case of the direct forecast method, interval forecasts for a given horizon are obtained 'directly,' based on the square root of the error variance of the direct regression. --- # Interval Forecast Evaluation Suppose `\(y_{t+h}: t=R,\ldots,T-h\)` is a sequence of `\(P\)` (observed) realizations of the random variable, where `\(P=T-R-h+1\)`. Suppose `\(\left\{l_{t+h|t,\pi},u_{t+h|t,\pi}\right\}\)` is a corresponding sequence of interval forecasts for some *ex ante* coverage probability, `\(\pi\)`. Obtain an indicator function, `\(I_t(h,\pi)\)` as follows: `$$I_t(h,\pi) = \left\{\begin{array} {ll} 1 & \text{if}~~y_{t+h}\in \left[l_{t+h|t,\pi},u_{t+h|t,\pi}\right]\\ 0 & \text{otherwise} \end{array}\right.$$` --- # Interval Forecast Evaluation Let `\(\tilde{\pi}=E\left[I_t(h,\pi)\right]\)` denote the *ex post* coverage, that is, the proportion of instances when the observed variable lies within the forecast interval. Suppose `\(P_1\)` and `\(P_0\)` are numbers of realizations that fall within and outside the forecast intervals, respectively. From the binomial distribution, the likelihood under the null hypothesis is `$$L(\pi)=(1-\pi)^{P_0}(\pi)^{P_1},$$` and the likelihood under the alternative alternative hypothesis is `$$L(\tilde{\pi})=(1-\tilde{\pi})^{P_0}(\tilde{\pi})^{P_1}.$$` --- # Interval Forecast Evaluation A test of correct unconditional coverage then is given by the likelihood ratio test statistic: `$$LR=-2\ln\left[\frac{L(\pi)}{L(\tilde{\pi})}\right],$$` which is `\(\chi^2\)` distributed with 1 degree of freedom. --- # Readings Marcellino, Stock & Watson (2006) Christoffersen (1998)