Calculating Interval Forecasts

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1 Calculating Chapter 7 (Chatfield) Monika Turyna & Thomas Hrdina Department of Economics, University of Vienna Summer Term 2009

2 Terminology An interval forecast consists of an upper and a lower limit between which a future value is expected to lie with a prescribed probability. Example: Inflation in the next quarter will lie in the interval [1%, 2.5%] with a 90% probability The limits are called forecast limits or prediction bounds while the interval is referred to as prediction interval (P.I.) Note: the term confidence interval usually applies to estimates of fixed but unknown parameter values while a P.I. is an estimate of an unknown future value of a random variable

3 Focus of Attention In what follows we concentrate on computing P.I.s for a single variable at a single (future) time point We do not cover the more difficult problem of calculating a P.I. for a single variable over a longer time horizon or a P.I. for different variables at the same time point Furthermore, we will cover a variety of approaches for calculating P.I.s the various methods for forecasting time-series require different approaches

4 Importance of P.I.s Predictions in form of point forecasts provide no guidance as to their likely accuracy P.I.s, in contrast, allow to assess future uncertainty, enable different strategies to be planned for different outcomes indicated by the P.I. and make a thorough comparison of forecasts from different methods possible

5 Some Reasons for Disregard Rather neglected topic in the statistical literature, i.e. textbooks and journal papers No generally accepted method of calculating P.I.s Theoretical P.I.s difficult or impossible to evaluate (e.g. for some multivariate or non-linear models) Properties of empirically based methods based on within-sample residuals have been little studied Software packages do not produce P.I.s at all or only on a limited scale

6 Density Forecasts Finding the entire probability distribution of a future value is called density forecasting For linear models with normally distributed innovations, the density forecast is usually a normal distribution with the mean equal to the point forecast and the variance equal to that used for computing the prediction interval. Of course, given the density forecast one can construct the P.I.s for any desired level of probability Problem: when forecast error distribution is not normal

7 Fan Charts Idea: for consecutive future values, construct several P.I.s for different probabilities (e.g. from 10% to 90%) and plot them in the same graph using different levels of shading for different probabilities Usually the darkest shade covers the P.I.s with a 10% probability while the lightest shade covers the P.I.s with a 90% probability Intervals typically get wider, indicating increasing uncertainty about future values, i.e. they fan out Fan charts could become a valuable tool for presenting the uncertainty attached to forecasts [... ] (Chatfield, 2000)

8 Density Forecasts & Fan Charts Example

9 Density Forecasts & Fan Charts Example (cont.)

10 Notation Conditional Forecast Error An observed time series (x 1, x 2,..., x N ) is regarded as a finite realisation of a stochastic process {X t } The point forecast of the random variable X N+h, conditional on data up to time N, is denoted by ˆX N (h) when regarded as a random variable and by ˆx N (h) when regarded as a particular value The conditional forecast error conditional on data up to time N and on the particular forecast is the random variable e N (h) = X N+h ˆx N (h) The observed value of e N (h) = x N+h ˆx N (h) only becomes available at time N + h

11 Notation Forecast Errors & Fitted Residuals It is important to differentiate between the (out-of-sample) conditional forecast errors e N (h), the fitted residuals (within-sample forecast errors) and the model innovations The out-of-sample observed forecasting errors e N (h) = x N+h ˆx N (h) are true forecasting errors The within-sample observed forecasting errors [x t ˆx t 1 (1)], for t = 2,..., N, are merely the residuals from the fitted model, i.e. the difference between observed and fitted values

12 Notation Forecast Errors & Fitted Residuals (cont.) These fitted residuals will not be the same as the true model innovations because they depend on parameter estimates (and perhaps on estimated starting values) They are also not true forecasting errors if the parameters have been estimated using data up to N However, if one finds the true model and the latter does not change then it is reasonable to expect the true forecast errors to have similar properties as the fitted residuals Practice: true forecast errors tend to be larger than expected from within-sample fit (change in the model?)

13 Prediction Mean Square Error (PMSE) The uncertainty of an h-step-ahead forecast of a single variable is assessed with its prediction mean square error given by E[e N (h) 2 ] For an unbiased forecast, i.e. where the conditional expectation E[X N+h ] = ˆx N (h), it holds that E[e N (h)] = 0 and thus E[e N (h) 2 ] = Var[e N (h)] Note: we are not interested in the variance of the forecast but in the variance of the forecast error (the particular value of the point forecast ˆx N (h) is determined exactly and has variance zero)

14 Prediction Mean Square Error (PMSE) Example Consider the zero mean AR(1) process X t = φ 1 X t 1 + ɛ t, where {ɛ t } are independent N(0, σ 2 ɛ ) Assuming that we know φ 1 and σ 2 ɛ, it can be shown that the true-model PMSE the true forecast error variance is given by E[e N (h) 2 ] = σ 2 ɛ (1 φ 2h 1 )/(1 φ 2 1) (1) In practice we would have to replace φ 1 and σ 2 ɛ with sample estimates

15 Prediction Mean Square Error (PMSE) Example (cont.) Assume that we want to calculate the PMSE of a one-step-ahead forecast, i.e. we want to know E[e N (1) 2 ] Since we do not know the true parameter values, the one-step-ahead forecasting error is given by e N (1) = X N+1 ˆx N (1) = φ 1 x N + ɛ N+1 ˆφ 1 x N = (φ 1 ˆφ 1 )x N + ɛ N+1 If the estimate ˆφ 1 is unbiased then we get E[e N (1) 2 ] = E[ɛ 2 N+1 ] = σ2 ɛ which is the same as for h = 1 in equation (1)

16 Prediction Mean Square Error (PMSE) Bias Correction The previous example shows that analysts have to bear in mind the effects of parameter uncertainty on estimates of the PMSE Though PMSE estimates can be corrected to allow for parameter uncertainty, the formulas are complex and corrections are merely of order 1/N Overall, the effect of parameter uncertainty seems likely to be of a smaller order of magnitude in general than that due to other sources, notably the effects of model uncertainty and the effects of errors and outliers [... ]. (Chatfield, 2000)

17 Introduction General formula for a 100(1 α)% P.I for X N+h ˆx N (h) ± z α/2 Var[eN (h)] Symmetric about ˆxN (h) assumes the forecast is unbiased E[e N (h) 2 ] = Var[e N (h)] Assumes errors are normally distributed [sometimes for short series z α/2 replaced by the respective percentage point of a t-distribution] The latter assumption usually violated even for a linear model with Gaussian innovations, when model parameters estimated from the same data used to compute forecasts For any method the main problem is to evaluate Var[e N (h)]

18 P.I.s from a Probability Model P.I.s derived from a probability model Ignore parameter uncertainty Example ARIMA forecasting: Xt = ε t + ψ 1 ε t 1 + ψ 2 ε t then: en (h) = [X N+h ˆX N+h ] = ε N+h + h 1 j=1 ψ jε N+h j thus: Var[eN (h)] = [1 + ψ ψ2 h 1 ]σ2 ε In practice replace ψ i and σε 2 with ˆψ i and ˆσ ε 2

19 P.I.s from a Probability Model P.I.s derived from a probability model cont d Similar formulas available for: Vector ARMA models, structural state space models and various regressions (the latter typically allow for parameter uncertainty and are conditional in the sense that they depend on the particular values of the explanatory variables from where a prediction is being made) Typically not available for: complicated simultaneous equation models, non linear models, ARCH and other stochastic volatility models P.I.s usually overoptimistic if the true model is not known and must be chosen from a class of models

20 Informal models P.I.s without model identification What to do when a forecasting method is selected without any formal model identification procedure? assume that the method is optimal (in some sense) apply some empirical procedure

21 Informal models P.I.s assuming a method is optimal Example: Exponential smoothing Optimal for ARIMA(0,1,1) PMSE formula: Var[e N (h)] = [1 + (h 1)α 2 ]σ 2 e where σ 2 e = Var[e n (1)] Should one use this formula even if the model has not been formally identified? Reasonable if: Observed one-step-ahead forecast errors show no obvious autocorrelation No other obvious features of the data (e.g. trend) which need to be modelled

22 Informal models Forecasting not based on a probability model Assume that the method is optimal in the sense that the one-step ahead errors are uncorrelated. Easy to check by looking at the correlogram of the one-step-ahead errors: if there is correlation we have more structure in the data which should improve the forecast. If we assume that one step ahead errors have also equal variance it should be possible to evaluate Var[e N (h)] in terms of Var[e N (1)] Example: Applied to Holt Winters method with additive and multiplicative seasonality; in the additive case found to be equivalent to results from SARIMA for which Holt Winters is optimal

23 Informal models Approximate formulae Used if no theoretical formula is available Sometimes, due to simplicity, used even if there exists a theoretical formula Usually very inaccurate Examples: 1 Var[e N (h)] = hσ 2 e where σ 2 e = Var[e N (1)] only true in a random walk model 2 Var[e N (h)] = ( h) 2 σ 2 e 3 Some formulas for the Holt Winters method [Bowermann and O Connel 1987]

24 Empirically based P.I.s Methods based on the observed distribution 1 Apply the forecasting method to all the past data Find the within sample forecast errors at 1,2,3... steps ahead from all available time origins Find the variance of these errors Assuming normality an approximate 100(1 α)%p.i. for XN+h is ˆx N (h) ± z α s e,h where s e,h is the standard deviation of the h steps ahead errors Values of s e,h can be unreliable for small N and large h

25 Empirically based P.I.s Methods based on the observed distribution cont d 2 Split the past data into two parts, fit the method to the first part Make predictions of the second part Resulting errors are much more like true forecast errors than those of the first method Refit the model with one additional observation in the first part and one less in the second and so on Interestingly it has been found that errors follow a gamma rather than normal distribution

26 Simulation and resampling methods Simulation Monte Carlo approach Given the probability time series model, simulate past and future behavior by generating a series of random variables Repeat many times to obtain a large set of outcomes, called pseudo data Evaluate P.I. by finding the interval within which the required percentage of future values lie Assumes the model has been correctly identified

27 Simulation and resampling methods Resampling Bootstrapping Sample from the empirical distribution of past fitted errors The procedure approximates the theoretical distribution of innovations by the empirical distribution of the observed residuals a distribution free approach Since in the time series context successive observations are correlated over time, thus bootstrapping over fitted errors [which are hopefully approximately independent] rather then independent observations However: procedure highly dependent on the fitted model choice

28 The Bayesian approach The Bayesian approach Given a suitable model, allows computation of a complete probability distribution for a future value Once probability distribution known, compute P.I.s by decision theoretic approach or Bayesian version of the P.I. general formula Alternatively: simulate the predictive distribution Bayesian model averaging Natural if the analyst relies not on a single model but on a mixture Use Bayesian methods to find a sensible set of models and to average over these in a appropriate way

29 P.I.s for transformed variables P.I.s for transformed variables Non linear transformation of variable X t : Y t = g(x t ) [e.g. logarithmic or Box Cox transformation] P.I.s for Y N+h can be calculated in an appropriate way But: How to get back P.I.s for the original variable? Usually point forecast for X N+h namely g 1 [ŷ N (h)]is biased since E[g 1 (Y )] g 1 [E(Y )] E.g.: If the predictive distribution of Y N+h is symmetric with mean ŷ N (h) then g 1 [ŷ N (h) is the median of X N+h Fortunately: If the P.I. for Y N+h has a probability (1 α) then the retransformed P.I. for X N+h will have the same probability Often P.I. for X N+h will be asymmetric

30 Why are P.I.s too narrow? Empirical evidence shows that out-of-sample forecast errors tend to be larger than model-fitted residuals, implying that more than 5% of future observations will fall outside a 95% P.I. on average The various reasons why P.I.s are too narrow in general include 1. parameter uncertainty 2. non-normally distributed innovations 3. identification of the wrong model 4. changes in the structure underlying the model

31 Why are P.I.s too narrow? (cont.) ad 1) This problem can be mitigated by using theoretical PSME formulae which account for parameter uncertainty by incorporating correction terms; corrections of this form seem to be negligible in comparison with those accounting for other sources of uncertainty ad 2) A common problem which is often associated with asymmetry or heavy tails; in particular the latter, which is due to outliers and errors in the data, can have severe effects on model identification and on the resulting point forecasts and P.I.s

32 Why are P.I.s too narrow? (cont.) ad 3) This problem is related to model uncertainty and can be mitigated by applying appropriate diagnostic checks (e.g. checking whether the one-step-ahead fitted residuals are uncorrelated and have constant variance; if this is not the case than there is more structure which should be exploited) ad 4) The underlying structure changes if it slowly evolves over time or there are sudden shifts; in both cases the model parameters will change (Chatfield argues that an observation that falls outside a P.I. could indicate a change in the underlying model)

33 Summary The basic message from this lecture is that P.I.s are a reasonable extension to point forecasts Wherever applicable, P.I.s should be calculated by formulating a model that approximates the DGP well and from which the PMSE can be used to obtain the interval To allow for parameter uncertainty correction terms can be used to calculate the PMSE; since the correction is rather small these terms are often omitted It is vital to note that the approach just described rests on the assumptions that the correct model has been fitted, the errors are normally distributed and that the future is like the past

34 Summary (cont.) The various approximate formulae for calculating P.I.s can be very inaccurate and should therefore not be used If theoretical formulae are not available or there is doubt about the assumptions stated above, empirically based resampling methods are a good alternative One should always bear in mind that P.I.s tend to be too narrow in practice, in particular because the wrong model has been identified or the model has changed Alternatively one could also consider different approaches to calculating P.I.s (e.g. Bayesian model averaging)

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