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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
• Nonlinear 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 + θ1ut1 + θ 2ut2 + θ 3ut3 + ut
yt+1 = μ + θ 1ut + θ 2ut1 + θ 3ut2 + ut+1
yt+2 = μ + θ 1ut+1 + θ 2ut + θ 3ut1 + 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 , yt1, ..., and ut , ut1
48 Forecasting with MA Models (cont’d) ft, 1 = E(yt+1  t ) =
= E(μ + θ 1ut + θ 2ut1 + θ 3ut2 + ut+1)
μ + θ 1ut + θ 2ut1 + θ 3ut2 ft, 2 = E(yt+2  t ) =
= E(μ + θ 1ut+1 + θ 2ut + θ 3ut1 + ut+2)
μ + θ 2ut + θ 3ut1 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 + φ1yt1 + φ 2yt2 + ut
yt+1 = μ + φ 1yt + φ 2yt1 + 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 2yt1 + ut+1)
= μ + φ 1E(yt) + φ 2E(yt1)
= μ + φ 1yt + φ 2yt1
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, s1 + φ 2 ft, s2 • 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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 Summer '13
 JaneBargers
 Financial Markets

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