Methods 1 assume no change fyts yt 2 forecasts

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Unformatted text preview: of its future value: E ( yt Ω t −1 ) = E (β1 + β 2 x2t + K + β k xkt + ut ) But what are = β1 + β 2 E ( x2t ) + K + β k E ( xkt ) etc.? We could use , so = 46 Models for Forecasting (cont’d) • Time Series Models The current value of a series, yt, is modelled as a function only of its previous values and the current value of an error term (and possibly previous values of the error term). • Models include: • simple unweighted averages • exponentially weighted averages • ARIMA models • Non-linear models – e.g. threshold models, GARCH models, etc. 47 Forecasting with MA Models • An MA(q) only has memory of q. e.g. say we have estimated an MA(3) model: yt = μ + θ1ut-1 + θ 2ut-2 + θ 3ut-3 + ut yt+1 = μ + θ 1ut + θ 2ut-1 + θ 3ut-2 + ut+1 yt+2 = μ + θ 1ut+1 + θ 2ut + θ 3ut-1 + ut+2 yt+3 = μ + θ 1ut+2 + θ 2ut+1 + θ 3ut + ut+3 • We are at time t and we want to forecast 1,2,..., s steps ahead. • We know yt , yt-1, ..., and ut , ut-1 48 Forecasting with MA Models (cont’d) ft, 1 = E(yt+1 | t ) = = E(μ + θ 1ut + θ 2ut-1 + θ 3ut-2 + ut+1) μ + θ 1ut + θ 2ut-1 + θ 3ut-2 ft, 2 = E(yt+2 | t ) = = E(μ + θ 1ut+1 + θ 2ut + θ 3ut-1 + ut+2) μ + θ 2ut + θ 3ut-1 ft, 3 = E(yt+3 | t ) = = E(μ + θ 1ut+2 + θ 2ut+1 + θ 3ut + ut+3) μ + θ 3ut ft, 4 = E(yt+4 | t ) = μ ft, s = E(yt+s | t ) = μ ∀s≥4 49 Forecasting with AR Models • Say we have estimated an AR(2) yt = μ + φ1yt-1 + φ 2yt-2 + ut yt+1 = μ + φ 1yt + φ 2yt-1 + ut+1 yt+2 = μ + φ 1yt+1 + φ 2yt + ut+2 yt+3 = μ + φ 1yt+2 + φ 2yt+1 + ut+3 ft, 1 = E(yt+1 | t ) = E(μ + φ 1yt + φ 2yt-1 + ut+1) = μ + φ 1E(yt) + φ 2E(yt-1) = μ + φ 1yt + φ 2yt-1 ft, 2 = E(yt+2 | t ) = E(μ + φ 1yt+1 + φ 2yt + ut+2) = μ + φ 1E(yt+1) + φ 2E(yt) = μ + φ 1 ft, 1 + φ 2yt 50 Forecasting with AR Models (cont’d) ft, 3 = E(yt+3 | t ) = E(μ + φ 1yt+2 + φ 2yt+1 + ut+3) = μ + φ 1E(yt+2) + φ 2E(yt+1) = μ + φ 1 f t, 2 + φ 2 f t, 1 • We can see immediately that ft, 4 = μ + φ 1 ft, 3 + φ 2 ft, 2 etc., so ft, s = μ + φ 1 ft, s-1 + φ 2 ft, s-2 • Can easily generate ARMA(p,q) forecasts in the same way. 51 How can we test whether a forecast is accurate or not? •For example, say we predict that tomorrow’s return on the FTSE will be 0.2, but the outcome is actually -0.4. Is this accurate? Define ft,s as the forecast made at time t for s steps ahead (i.e. the forecast made for time t+s), and yt+s as the realised value of y at time t+s. • Some of the most popular criteria for assessing the accuracy of time series forecasting techniques are: MAE is given by Mean absolute percentage error: 52 How can we test whether a forecast is accurate or not? (cont’d) • It has, however, also recently been shown (Gerlow et al., 1993) that the accuracy of forecasts according to traditional statistical criteria are not related to trading profitability. • A measure more closely correlated with profitability: % correct sign predictions = where zt+s = 1 if (xt+s . ft,s ) > 0 zt+s = 0 otherwise 53 Forecast Evaluation Example • Given the following forecast and actual values, calculate the MSE, MAE and percentage of correct sign predictions: • MSE = 0.079, MAE = 0.180, % of correct sign predictions = 40 54...
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