Econ 240C Lecture Thirteen 1 5-15-2003I. Exponential Smoothing is a useful forecasting technique for small samples of data where it would be difficult to estimate a Box-JenkinsARIMA model. Exponential smoothing is based on recursive formulas and hence requires initial conditions. It could be undertaken with pencil and paper, or a spread sheet, but is less tedious with Eviews or TSP. II. Simple (Single) exponential smoothing relates the forecast time series f(t) to the observed time series, y(t) in the following recursive formula: f(t) = a y(t-1) + (1-a) f(t-1) ,where a is a parameter which can either be chosen or estimated. A way to initiate the formula is to set the first observation, y(1), equal tothe first forecast. f(1), and iterate from there,f(1) = y(1),f(2) = a y(1) + (1-a) f(1) = y(1)f(3) = a y(2) + (1-a) f(2) = a y(2) + (1-a) y(1)etc.The forecast error is equal to the observed minus the forecast:e(t) = y(t) - f(t),e(1) = y(1) - f(1) = y(1) - y(1) =0e(2) = y(2) - f(2) = y(2) - y(1),etc. The sum of squared errors depends upon the choice of the parameter a, [e(t)]2 = [y(t) -f(t)]2so a can be chosen to minimize this sum, using a grid search or gradient search algorithm.Econ 240C Lecture Thirteen 2 5-15-2003III. The forecasting formula can be written using the lag operator whichshows that the forecast is a geometrically distributed lag of the past observations:f(t+1) = a y(t) + (1-a) f(t)f(t+1) = a y(t) + (1-a)Z f(t+1),[1-(1-a)Z] f(t+1) = a y(t),f(t+1) = {a/[1-(1-a)Z]}*y(t)f(t+1) = a[1+(1-a)Z+(1-a)2Z2+(1-a)3Z3+...]y(t)f(t+1) = a y(t) + a(1-a)y(t-1) + a(1-a)2y(t-2) + ...where the weights, w(i) at lags zero, one, two, etc. sum to one:w(i) = a + a(1-a) + a(1-a)2+... = a[1+(1-a)+(1-a)2+...]w(i) = a/[1-(1-a)] = a/a = 1.Thus the weights have the same properties as the discrete density function, p(i), for a geometrically distributed random variable.:w(0) = a = p(0)w(1) = a(1-a) = p(1)w(2) = a(1-a)2 = p(2)...w(i) = a(1-a)i = p(i).The probability generating function, (Z)of a random variable taking discrete integral values, i, with probability density, p(i), is defined as:(Z)= p(0)Z0 + p(1)Z1 + p(2)Z2 + ... .Econ 240C Lecture Thirteen 3 5-15-2003For the geometric distribution function where p(i) = a(1-a)i ,(Z)= a + a(1-a)Z + a(1-a)2Z2 + ... = a/[1-(1-a)Z]i.e. the probability generating function has the same form as the impulse response function relating the forecast, f(t+1), to the observations, y(t).The derivative of the probability generating function, evaluated at Z equals one, is the mean or expected value of the random variable, and so for the geometric will provide the mean over the integral values, i, this discrete random variable can take, which equivalently will be the mean lag of our distributed lag since the weights, w(i), equal the probability density, p(i), for the geometric. Taking the derivative(applying the quotient rule):d(Z)/dZ = a(1-a)/[1-(1-a)Z]2d(Z=1)/dZ = a(1-a)/[1-(1-a)]2 =a(1-a)/a2 = (1-a)/a.Thus the smoothing parameter a, is related to the average lag.IV. Simple Exponential Smoothing as an ARIMA ProcessThe recursive formula,f(t) = a y(t-1) + (1-a) f(t-1),and the definition of the forecast error,e(t) = y(t) - f(t)provide two equations in three variables. One of these equations can be used to eliminate one of the variables leavingan equation relating the other two. For example, the forecast f(t) can be eliminated to show the presumed ARIMA structure for y(t). From the error definition,f(t) = y(t) - e(t),Econ 240C Lecture Thirteen 4 5-15-2003and lagging by one,f(t-1) = y(t-1) - e(t-1),and substituting in the recursive formula for f(t) and f(t-1): y(t) - e(t) = a y(t-1) + (1-a)[y(t-1) - e(t-1)],and rearranging terms:y(t) = y(t-1) + e(t) - (1-a) e(t-1),or y(t) - y(t-1) = e(t) - (1-a) e(t-1),i.e. the first difference in y(t) is a first order moving average process, i.e. y(t) is (0,1,1) where this notation is for (p,d,q) and p is the order of the autoregressive part or polynomial in lag(Z)AR(p), d is the order of differencing, and q is the order of the moving average part or polynomial in lag(Z) or MA(q).If instead of substituting out for the forecast we substitute out for the observed series, we obtain a recursive formula relating the forecast to an update based on the random shock:f(t) = a[f(t-1) + e(t-1)] + (1-a)f(t-1),and rearranging terms:f(t) = f(t-1) + a e(t-1),i.e. the forecast is last period’s forecast plus a fraction, a, of the new information or random shock, e(t-1).V. Double Exponential SmoothingEcon 240C Lecture Thirteen 5 5-15-2003Double exponential smoothing constructs the forecast from a constructed series "levels MEAN", L(t), and the "trend", R(t), and is used for forecasting trended series or near-random walks. With double exponential smoothing it is possible to forecast ahead more than one period:f(t+k) = L(t) + k R(t), k≥1L(t) = a y(t) + (1-a)[L(t-1) + R(t-1)]R(t) = b [L(t) - L(t-1)] + (1-b) R(t-1)In TSP, the program sets a = b for double exponential smoothing and allows a and b to differ in the Holt-Winters without a seasonal term, so that the three equations above cover both of these cases.VI. Holt-Winters with an Additive Seasonal TermThe Holt-Winters procedure for smoothing can handle seasonal components as well as trend components in a time series. This set of four equations illustrates a seasonal term formonthly data:f(t + k) = L(t) + k R(t) + S(t+k-12), k≥1L(t) = a [y(t) - S(t-12)] + (1-a)[L(t-1) + R(t-1)]R(t) = b [L(t) - L(t-1)] + (1-b) R(t-1)S(t) = c{y(t) - L(t)] + (1-c) S(t-12)Econ 240C Lecture Thirteen 6
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