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Interval Forecasting Introduction Prediction Mean Square Error Prediction Intervals Empirically Based P.I.s Summary Interval Forecasting Based on Chapter 7 of the Time Series Forecasting by Chatfield Econometric Forecasting, January 2008 Pekalski, Swierczyna, Zalewski interval forecasting Introduction Prediction Mean Square Error Prediction Intervals Empirically Based P.I.s Summary Outline 1 Introduction 2 Prediction Mean Square Error 3 Prediction Intervals 4 Empirically Based P.I.s 5 Summary Pekalski, Swierczyna, Zalewski interval forecasting Introduction Introduction Prediction Mean Square Error Terminology Prediction Intervals Interval Forecasts Empirically Based P.I.s Density Forecast Summary Fan Chart Introduction Introduction Most forecasters realize the importance of providing interval forecasts and point forecasts in order to: asses future uncertainty, enable different strategies to be planned for the range of possible outcomes indicated by the interval forecast, compare forecasts from different methods more thoroughly. Pekalski, Swierczyna, Zalewski interval forecasting Introduction Introduction Prediction Mean Square Error Terminology Prediction Intervals Interval Forecasts Empirically Based P.I.s Density Forecast Summary Fan Chart Introduction Terminology An interval forecast usually consists of an upper and lower limit. The limits are called forecast limits, prediction bounds. The interval is called a confidence interval, forecast region, prediction interval. Pekalski, Swierczyna, Zalewski interval forecasting Introduction Introduction Prediction Mean Square Error Terminology Prediction Intervals Interval Forecasts Empirically Based P.I.s Density Forecast Summary Fan Chart Introduction Predictions Predictions: are not often a prediction intervals, most often are given as a single value. Pekalski, Swierczyna, Zalewski interval forecasting Introduction Introduction Prediction Mean Square Error Terminology Prediction Intervals Interval Forecasts Empirically Based P.I.s Density Forecast Summary Fan Chart Introduction Interval Forecasts Reasons for not using interval forecasts: rather neglected in statistical literature, no generally accepted method for calculating prediction intervals (with some exception), theoretical prediction intervals are difficult or impossible to evaluate for many econometric models containing many equations or which depend on non-linear relationship. Pekalski, Swierczyna, Zalewski interval forecasting Introduction Introduction Prediction Mean Square Error Terminology Prediction Intervals Interval Forecasts Empirically Based P.I.s Density Forecast Summary Fan Chart Introduction Density Forecast By density forecast we mean finding the entire probability distribution of a future value of interest. Linear models with normally distributed innovations: the density forecast is typically normal with mean equal to the point forecast and variance equal to that used in computing a predicion interval. Linear model without normally distributed innovations: seems to be more prevalent and used e.g. in forecasting volatility, different percentiles or quantiles of the conditional probability distribution of future values are estimated. Pekalski, Swierczyna, Zalewski interval forecasting Introduction Introduction Prediction Mean Square Error Terminology Prediction Intervals Interval Forecasts Empirically Based P.I.s Density Forecast Summary Fan Chart Introduction Fan Chart Fan charts Pekalski, Swierczyna, Zalewski interval forecasting Formula Introduction Potential Pitfall Prediction Mean Square Error Example Prediction Intervals Known vs. Unknown Parameters Empirically Based P.I.s Conditioning Forecast Error Summary Example Prediction Mean Square Error Preditcion Mean Square Error (PMSE): 2 E [en(h) ]. Forecast is unbiased that is, when xˆN (h) is the mean of the predictive distriburion, then 2 E [eN (h)] = 0 and E [eN (h) ] = Var[eN (h)]. Pekalski, Swierczyna, Zalewski interval forecasting Formula Introduction Potential Pitfall Prediction Mean Square Error Example Prediction Intervals Known vs. Unknown Parameters Empirically Based P.I.s Conditioning Forecast Error Summary Example Prediction Mean Square Error Potential Pitfall Assesing forecast uncertainty, remember: the var of the forecast 6= the var of the forecast error. Given data up to time N and a particular method or model: the forecast xˆN (h) is not a random variable, it has variance of zero, XN+h and eN (h) are random variables, condtioned by the observed data. Pekalski, Swierczyna, Zalewski interval forecasting Formula Introduction Potential Pitfall Prediction Mean Square Error Example Prediction Intervals Known vs. Unknown Parameters Empirically Based P.I.s Conditioning Forecast Error Summary Example Prediction Mean Square Error How to evaluate: 2 E [eN (h) ] or Var[eN (h)] and what assumptions should be made? Pekalski, Swierczyna, Zalewski interval forecasting Formula Introduction Potential Pitfall Prediction Mean Square Error Example Prediction Intervals Known vs. Unknown Parameters Empirically Based P.I.s Conditioning Forecast Error Summary Example Prediction Mean Square Error Example Consider the zero-mean AR(1): 2 Xt = αXt−1 + εt , {εt } ∼ N(0, σε ). 2 Assume: complete knowledge of the model (α and σε ). The point forecasts xˆN (h) will be h h αˆ xN rather than α xN . Pekalski, Swierczyna, Zalewski interval forecasting Formula Introduction Potential Pitfall Prediction Mean Square Error Example Prediction Intervals Known vs. Unknown Parameters Empirically Based P.I.s Conditioning Forecast Error Summary Example Prediction Mean Square Error Bias of PMSE Even assuming that the true model is known a priori, there will still be biases in the usual estimate obtained by substituting sample estimates of the model parameters and the residual variance into the true-model PMSE formula. Pekalski, Swierczyna, Zalewski interval forecasting Formula Introduction Potential Pitfall Prediction Mean Square Error Example Prediction Intervals Known vs. Unknown Parameters Empirically Based P.I.s Conditioning Forecast Error Summary Example Known vs. Unknown Parameters Consider the case of h = 1 and conditioning on Xn = xn, eN (1) = XN+1 − xˆN (1). If the model parameters were known, then xˆN (1) = αxN ⇒ eN (1) = εN+1 If parameters are not known: eN (1) = XN+1 − xˆN (1) = αxN + εN+1 − αˆxN = (α − αˆ)xN + εN+1 Assume: parameters estimates are obtained by a procedure that is asymptotically equivalent to maximum likelihood. Pekalski, Swierczyna, Zalewski interval forecasting Formula Introduction Potential Pitfall Prediction Mean Square Error Example Prediction Intervals Known vs. Unknown Parameters Empirically Based P.I.s Conditioning Forecast Error Summary Example Prediction Mean Square Error Conditional and Unconditional Errors Looking once more at equation eN (1) = (α − αˆ)xN + εN+1. Consider: conditional on xN forecast error: xN is fixed αˆ - biased estimator of α, then the expected value of eN (1) need not be zero. unconditional forecast error: If, however, we average over all possible values of xN , as well as over εN , then it can be shown that te expectation will indeed be zero giving an unbiased forecast. Pekalski, Swierczyna, Zalewski interval forecasting Formula Introduction Potential Pitfall Prediction Mean Square Error Example Prediction Intervals Known vs. Unknown Parameters Empirically Based P.I.s Conditioning Forecast Error Summary Example Prediction Mean Square Error Computing PMSE Important when computing PMSE. To have unconditional PMSE: average over the distribution of future innovations (e.g. eN+1), average over the distribution of current observed values (e.g. xN ). eN (1) = (α − αˆ)xN + eN+1 Pekalski, Swierczyna, Zalewski interval forecasting Formula Introduction Potential Pitfall Prediction Mean Square Error Example Prediction Intervals Known vs. Unknown Parameters Empirically Based P.I.s Conditioning Forecast Error Summary Example Prediction Mean Square Error Unconditional PMSE Unconditional PMSE can be usefull to assess the ’success’ of a forecasting method on average. This apporach if used to compute P.I.s, it effectively assumes that te observatoins used to estimate the model parameters are independent of those used to construct the forecasts. This assumption can be justified asymptotically. Box assesed that the correction terms 1 would generally be of order N (effect of parameter uncertainty). Pekalski, Swierczyna, Zalewski interval forecasting Formula Introduction Potential Pitfall Prediction Mean Square Error Example Prediction Intervals Known vs. Unknown Parameters Empirically Based P.I.s Conditioning Forecast Error Summary Example Parameters Uncertainty Correction Assume: K-variable vector AR(p) process. True model PMSE at lead time one has to be multiplied by the correction factor 1 1 [1 + K N ] + o( N ) to give the corresponding unconditional PMSE allowing for parameters uncertainty. What does it mean? the more parameters (K ), the shorter the series (N), ⇒ the greater will be the correction term. Pekalski, Swierczyna, Zalewski interval forecasting Formula Introduction Potential Pitfall Prediction Mean Square Error Example Prediction Intervals Known vs. Unknown Parameters Empirically Based P.I.s Conditioning Forecast Error Summary Example Parameters Uncertainty Example N = 50, K = 1 and p = 2 the correction for the square root of PMSE is only 2%. for N = 30, K = 3 and p = 2 the correction for the square root of PMSE rises to 6%. Pekalski, Swierczyna, Zalewski interval forecasting Formula Introduction Potential Pitfall Prediction Mean Square Error Example
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