# 3. Regression & Exponential Smoothing

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1 3. Regression & Exponential Smoothing 3.1 Forecasting a Single Time Series Two main approaches are traditionally used to model a single time series z 1, z 2,..., z n 1. Models the observation z t as a function of time as z t = f(t, β) + ε t where f(t, β) is a function of time t and unknown coefficients β, and ε t are uncorrelated errors. 1

2 Examples: The constant mean model: z t = β + ε t The linear trend model: z t = β 0 + β 1 t + ε t Trigonometric models for seasonal time series z t = β 0 + β 1 sin 2π 12 t + β 2 cos 2π 12 t + ε t 2. A time Series modeling approach, the observation at time t is modeled as a linear combination of previous observations Examples: The autoregressive model: z t = j 1 π jz t j + ε t The autoregressive and moving average model: z t = p π j z t j + q θ i ε t i + ε t j=1 j=1 2

3 Discounted least squares/general exponential smoothing n w t [z t f(t, β)] 2 t=1 Ordinary least squares: w t = 1. w t = w n t, discount factor w determines how fast information from previous observations is discounted. Single, double and triple exponential smoothing procedures The constant mean model The Linear trend model The quadratic model 3

4 3.2 Constant Mean Model z t = β + ε t β: a constant mean level ε t : a sequence of uncorrelated errors with constant variance σ 2. If β, σ are known, the minimum mean square error forecast of a future observation at time n + l, z n+l = β + ε n+l is given by z n (l) = β E[z n+l z n (l)] = 0, E[z n+1 z n (l)] 2 = E[ε 2 t] = σ 2 100(1 λ) percent prediction intervals for a future realization are given by [β µ λ/2 σ; β + µ λ/2 σ] where µ λ/2 distribution. is the 100(1 λ/2) percentage point of standard normal 4

5 If β, σ are unknown, we use the least square estimate ˆβ to replace β n ˆβ = z = 1 n t=1 z t The l-step-ahead forecast of z n+l from time origin n by ẑ n (l) = z, ˆσ 2 = 1 n 1 n (z t z) 2 E[z n+l ẑ n (l)] = 0, Eˆσ 2 = σ 2, E[z n+1 ẑ n (l)] 2 = σ ( ) n 100(1 λ) percent prediction intervals for a future realization are given by [ ( β t λ/2 (n 1)ˆσ 1 + n)1 1 2 ( ] ; β + tλ/2 (n 1)ˆσ )1 2 n where t λ/2 (n 1) is the 100(1 λ/2) percentage point of t distribution with n 1 degree of freedom. t=1 5

6 Updating Forecasts ẑ n+1 = = 1 n + 1 (z 1 + z z n + z n+1 ) = 1 n + 1 [z n+1 + nẑ n (1)] n n + 1ẑn(1) + 1 n + 1 z n+1 = ẑ n (1) + 1 n + 1 (z n+1 ẑ n (1)) Checking the adequacy of the model Calculate the sample autocorrelation r k of the residuals z t z r k = n t=k+1 (z t z)(z t k z) n, k = 1, 2,... (z t z) 2 t=1 If n r k > 2: something might have gone wrong. 6

7 Example 3.1 Annual U.S. Lumber Production Consider the annual U.S. lumber production from 1947 through The data were obtained from U.S. Department of Commerce Survey of Current Business. The 30 observations are listed in Table Table 3.1: Annual Total U.S. Lumber Production (Millions of Broad Feet), (Table reads from left to right) 35,404 36,762 32,901 38,902 37,515 37,462 36,742 36,356 37,858 38,629 32,901 33,385 32,926 32,926 32,019 33,178 34,171 35,697 35,697 35,710 34,449 36,124 34,548 34,548 36,693 38,044 38,658 32,087 32,087 37,153 7

8 Figure 3.1: Annual U.S. lumber production from 1947 to 1976(in millions of board feet) x 104 Lumber production (MMbf) Year 8

9 The plot of the data in Figure 3.1. The sample mean and the sample standard deviation are given by z = 35, 625, ˆσ = { 1 29 (zt z) 2 } 1 2 = 2037 The sample auto correlations of the observations are list below Lag k Sample autocorrelation Comparing the sample autocorrelations with their standard error 1/ 30 =.18, we cannot find enough evidence to reject the assumption of uncorrelated error terms. 9

10 The forecast from the constant mean model are the same for all forecast lead times and are given by ẑ 1976 (l) = z = 35, 652 The standard error of these forecast is given by ˆσ 1 + 1/n = 2071 A 95 percent prediction interval is given by [35, 652 ± (2.045)(2071)] or [31, 417, 39, 887] If new observation become available, the forecasts are easily updated. For example if Lumber production in 1977 was 37,520 million board feet. Then the revised forecasts are given by ẑ 1977 (l) = ẑ 1976 (1) + 1 n + 1 [z 1977 ẑ 1976 (1)] = 35, [37, , 652] 31 = 35,

11 3.3 Locally Constant Mean Model and Simple Exponential Smoothing Reason: In many instances, the assumption of a time constant mean is restrictive. It is more reasonable to allow for a mean that moves slowly over time Method: Give more weight to the most recent observation and less to the observations in the distant past ẑ n (l) = c n 1 t=0 w t z n t = c[z n + wz n w n 1 z 1 ] w( w < 1): discount coefficient, c = (1 w)/(1 w n ) is needed to normalized sum of weights to 1. If n and w < 1, then w n 0, then ẑ n (l) = (1 w) j 0 w j z n j 11

12 Smoothing constant: α = 1 w. Smoothing statistics S n = S [1] n = (1 w)[z n + wz n 1 + w 2 z n 2 + ] = α[z n + (1 α)z n 1 + (1 α) 2 z n 2 + ] Updating Forecasts: (As easy as the constant mean model) S n = (1 w)z n + ws n 1 = S n 1 + (1 w)[z n S n 1 ] ẑ n (1) = (1 w)z n + wẑ n 1 (1) = ẑ n 1 (1) + (1 w)[z n ẑ n 1 (1)] 12

13 Actual Implementation of Simple Exp. Smoothing Initial value for S 0 S n = (1 w)[z n + wz n w n 1 z 1 ] + w n S 0 1. S 0 = z, (mean change slowly, α. = 0); 2. S 0 = z 1, (local mean changes quickly α. = 1); 3. Backforecast S j = (1 w)z j + ws j+1, S n+1 = z n, S 0 = z 0 = S1 = (1 w)z 1 + ws2 Choice of the Smoothing Constant: α = 1 w e t 1 (1) = z t ẑ t 1 (1) = z t S t 1, (one step ahead forecast error). Then minimize n SSE(α) = e 2 t 1(1). t=1 13

14 The smoothing constant that is obtained by simulation depends on the value of S 0 Ideally, since the choice of α depend on S 0, one should choose α and S 0 jointly Examples If α = 0, one should choose S 0 = z. If α = 1, one should choose S 0 = z 1 If 0 < α < 1, one could choose S 0 as the backforecast value: S 0 = α[z 1 + (1 α)z (1 α) n 2 z n 1 ] + (1 α) n 1 z n. 14

15 Example: Quarterly Iowa Nonfarm Income As an example, we consider the quarterly Iowa nonfarm income for The data exhibit exponential growth. Instead of analyzing and forecasting the original series, we first model the quarterly growth rates of nonfarm income. z t = I t+1 I t I t log I t+1 I t The constant mean model would be clearly inappropriate. Compared with the standard error 1/ 127 =.089, most autocorrelations are significantly different from zero Table 3.2: Sample Autocorrelations r k of Growth Rates of Iowa Nofarm Income (n=127) Lag k Sample autocorrelation r k

16 Iowa nonfarm income, first quarter 1948 to fourth quarter Iowa nofarm income (million \$) Year 16

17 Growth rates of Iowa nonfarm income, second quarter 1948 to fourth quarter Growth rate (%) Year 17

18 Since the mean is slowly changing, simple exponential smoothing appears to be an appropriate method. α = 0.11 and S 0 = z = SSE(.11) = n e 2 t 1(1) = ( 1.329) 2 + (.967) (.458) 2 + (.342) 2 = t=1 As a diagnostic check, we calculate the sample autocorrelations of the one-step-ahead forecast errors r k = n 1 t=k [e t (1) ē][e t k (1) ē] n 1 t=0 [e t (1) ē] 2, ē = 1 n n 1 t=0 e t (1). To assess the significance of the mean of the forecast errors, we compare it with standard error s/n 1/2 (1/ 127 =.089), where s 2 = 1 n n 1 t=0[e t (1) ē] 2 18

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23 3.4 Regression Models with Time as Independent Variable z n+j = m β i f i (j) + ε n+j = f (j)β + ε n+j i=1 f(j + 1) = Lf(j), L = (l ij ) m m full rank. (Difference equations). Equivalent model: z n+j = m βi f i (n + j) + ε n+j = f (n + j)β + ε n+j ; i=1 f(n + j) = L n f(j) β = L n β. Examples: Constant Mean Model, Linear Trend Model, Quadratic Trend Model, k th order Polynomial Trend Model, 12-point Sinusoidal Model 23

24 Estimation: ˆβ n minimizes n j=1 [z j f (j n)β] 2 y = (z 1, z 2,..., z n ), X = (f( n + 1),..., f(0)) Prediction X X = n 1 j=0 f( j)f ( j) ˆ=F n, X y = ˆβ n = F 1 n h n. n 1 j=0 f( j)z n j ˆ=h n ẑ n (l) = f (l)ˆβ n, Var(e n (l)) = σ 2 [1 + f (l)f 1 n f(l)], ˆσ 2 = 1 n m n 1 (z n j f ( j)ˆβ n ) 2. j=0 100(1 λ)% CI : ẑ n (l) ± t λ/2 (n m)ˆσ[1 + f (l)f 1 f(l)]

25 Updating Estimates and Forecasts: ˆβ n+1 = F 1 n+1 h n+1. F n+1 = F n + f( n)f ( n); h n+1 = n f( j)z n+1 j = f(0)z n+1 + j=0 = f(0)z n+1 + n 1 j=0 n 1 j=0 f( j 1)z n j L 1 f( j)z n j = f(0)z n+1 + L 1 h n ẑ n+1 (l) = f (l)ˆβ n+1. 25

26 3.5 Discounted Least Square and General Exponential Smoothing In discounted least squares or general exponential smoothing, the parameter estimates are determined by minimizing n 1 j=0 w j [z n j f ( j)β] 2 The constant w( w < 1) is a discount factor the discount past observation exponentially. Define W = diag(w n 1 w n 2 w 1); F n ˆ=X WX = h n ˆ=X Wy = n 1 j=0 n 1 j=0 w j f( j)f ( j) w j f( j)z n j 26

27 Estimation and Forecasts ˆβ n = F 1 n h n, ẑ n (l) = f (l)ˆβ n. Updating Parameter Estimates and Forecasts ˆβ n+1 = F 1 n+1 h n+1, F n+1 = F n + f( n)f ( n)w n ; h n+1 = n w j f( j)z n+1 j = f(0)z n+1 + wl 1 h n j=0 If n, then w n f( n)f ( n) 0, and F n+1 = F n + f( n)f ( n)w n F as n. Hence ˆβ n+1 = F 1 f(0)z n+1 + [L F 1 f(0)f (0)L ]ˆβ n = L ˆβn + F 1 f(0)[z n+1 ẑ n (1)]. ẑ n+1 (l) = f (l)ˆβ n+1. 27

28 3.5 Locally Constant Linear Trend and Double Exp. Smoothing Locally Constant Linear Trend Model z n+j = β 0 + β 1 j + ε n+j [ ] Definition f(j) = [1 j] 1 0, L =. 1 1 F = [ w j f( j)f w j ] jw ( j) = j jw j j 2 w j = [ 1 1 w w (1 w) 2 w (1 w) 2 w(1+w) (1 w) 2 ] Discount least squares that minimizing n 1 ˆβ n = F 1 h n = j=1 [ 1 w 2 (1 w) 2 (1 w) 2 (1 w) 3 w w j [z n j f ( j)β] 2, thus ] [ ] w j z n j jw j z n j 28

29 Thus ˆβ 0,n = (1 w 2 ) w j z n j (1 w) 2 jw j z n j ˆβ 1,n = (1 w) 2 w j z n j (1 w)3 w jw j z n j In terms of smoothing S [1] n = (1 w)z n + ws [1] n 1 = (1 w) w j z n j, S [2] n = (1 w)s [1] n + ws [2] n 1 = (1 w)2 (j + 1)w j z n j, S n [k] = (1 w)s n [k 1] + ws [k] n 1. (S [0] n = (no smoothing) = z n ) Then ˆβ 0,n = 2S [1] n S [2] n, ˆβ 1,n = 1 w w (S[1] n S n [2] ). ẑ n (l) = ˆβ 0,n + ˆβ 1,n l = (2 + 1 w w l)s[1] n (1 + 1 w w l)s[2] n. 29

30 Updating: ˆβ 0,n+1 = ˆβ 0,n + ˆβ 1,n + (1 w 2 )[z n+1 ẑ n (1)], ˆβ 1,n+1 = ˆβ 1,n + (1 w) 2 [z n+1 ẑ n (1)]; Or in another combination form: ˆβ 0,n+1 = (1 w 2 )z n+1 + w 2 ( ˆβ 0,n + ˆβ 1,n ), ˆβ 1,n+1 = 1 w 1 + w ( ˆβ 0,n+1 ˆβ 0,n ) + 2w 1 + w ˆβ 1,n. z n+1 ẑ n (1) = 1 1 w 2( ˆβ 0,n+1 ˆβ 0,n ˆβ 1,n ). 30

31 Implementation Initial Values for S [1] 0 and S [2] 0 S [1] 0 = ˆβ 0,0 w 1 w ˆβ 1,0, S [2] 1 = ˆβ 0,0 2w 1 w ˆβ 1,0 ; where the ( ˆβ 0,0, ˆβ 1,0 ) are usually obtained by considering a subset of the data fitted by the standard model z t = β 0 + β 1 t + ε t. Choice of the Smoothing Constant α = 1 w The smoothing constant α is chosen to minimize the SSE: SSE(α) = (z t ẑ t 1 ) 2 = [ z t ( 2 + α ) ( S [1] t 1 1 α α ) 2 S [2] 1 α t 1]. 31

32 Example: Weekly Thermostat Sales As an example for double exponential smoothing, we analyze a sequence of 52 weekly sales observations. The data are listed in Table and plotted in Figure. The data indicates an upward trend in the thermostat sales. This trend, however, does not appear to be constant but seems to change over time. A constant linear trend model would therefore not be appropriate. Case Study II: University of Iowa Student Enrollments As another example, we consider the annual student enrollment (fall and spring semester combined) at the University of Iowa. Observations for last 29 years (1951/1952 through 1979/1980) are summarized in Table. A plot of the observations is given in Figure. 32

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34 Thermostat Sales Week Weekly thermostat sales 34

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39 3.6 Regression and Exponential Smoothing Methods to Forecast Seasonal Time Series Seasonal Series: Series that contain seasonal components are quite common, especially in economics, business, and the nature sciences. Much of seasonality can be explained on the basis of physical reasons. The earth s rotation around the sun, for example, introduces a yearly seasonal pattern into may of the meteorological variables. The seasonal pattern in certain variables, such as the one in meteorological variables, is usually quite stable and deterministic and repeats itself year after year. The seasonal pattern in business and economic series, however, is frequently stochastic and changes with time. Apart from a seasonal component, we observe in many series an additional trend component. 39

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43 z t = T t S t + ε t 43 The traditional approach to modeling seasonal data is to decompose the series into three components: a trend T t, a seasonal component S t and an irregular (or error) component ε t The additive decomposition approach z t = T t + S t + ε t The multiplicative decomposition approach z t = T t S t ε t or log z t = Tt + St + ε t The other multiplicative model

44 3.6.1 Globally Constant Seasonal Models Consider the additive decomposition model z t = T t + S t + ε t Traditionally the trend component T t is modeled by low-order polynomials of time t: k t i T t = β 0 + β i i! The seasonal component S t can be described by seasonal indicators s S t = δ i IND ti i=1 where IND ti = 1 if t corresponds to the seasonal period i, and 0 otherwise, or by trigonometric functions m S t = A i sin( 2πi s t + φ i). i=1 where A i and φ i are amplitude and the phase shift of the sine function with frequency f i = 2πi/s. 44 i=1

45 Modeling the Additive Seasonality with Seasonal Indicators z t = β 0 + k i=1 β i t i i! + s i=1 δ i IND ti + ε t. Since it uses s + 1 parameters ( β 0 and s seasonal indicators) to model s seasonal intercepts, restrictions have to be imposed before the parameters can be estimated. Several equivalent parameterizations are possible Omit the intercept: β 0 = 0. Restrict s i=1 δ i = 0. Set one of the δ s equal to zero; for example δ s = 0. Mathematically these modified models are equivalent, but for convenience we usually choose (3). 45

46 Now we have the standard regression model: k t i s 1 z t = β 0 + β i i! + δ i IND ti + ε t. Hence i=1 i=1 β = (β 0, β 1,..., β k, δ 1,..., δ s 1 ); ˆβ = (XX ) 1 X y, y = (z 1, z 2,..., z n ); X is an n (k + s)matrix with tth row given by ) f t22 tk (t) = (1, t,,, k!, IND t1,, IND t,s 1 The minimum mean square error forecast of z n+l can be calculated from 100(1 α)% prediction interval ẑ n (l) = f (n + l)ˆβ where ẑ n (l) ± t λ/2 (n k s)ˆσ[1 + f (n + l)(xx ) 1 f(n + l)] 2 1, ˆσ 2 1 n = (z t f (t)ˆβ) n k s t=1

47 Change of Time Origin in the Seasonal Indicator Model As we mentioned in previous sections, it might be easier to update the estimates/ prediction if we use the last observational time n as the time origin. In this case k j i z n+j = β 0 + β i i! + s 1 δ i IND ji + ε n+j. i=1 i=1 ) f (j) = (1, j, j2 2,, jk k!, IND j1,, IND j,s 1 F n = Hence the prediction: n 1 j=0 100(1 α)% prediction interval ˆβ n = F 1 n h n f( j)f ( j),, h n = ẑ n (l) = f (l)ˆβ n 1 j=0 f( j)z n j. ẑ n (l) ± t λ/2 (n k s)ˆσ[1 + f (l)f 1 n f(l)] 1 2, 47

48 It can be shown that the forecast or fitting functions follow the difference equation f(j) = Lf(j 1), where L is a (k + s) (k + s) transition matrix L = [ ] L11 0 L 21 L 22 As an illustration, let us consider a model with quadratic trend and seasonal period s = 4 j 2 3 z n+j = β 0 + β 1 + β δ i IND ji + ε n+j Then the transition matrix L and the initial vector f(0) are given by i=1 L 11 = /2 1 1 L 21 = L 22 = Successive application of the difference equation f(j) = Lf(j 1) leads to f(j) = (1, j, j 2 /2, IND j1, IND j2, IND j3 ). 48

49 Modeling the Seasonality with Trigonometric Functions z t = T t + S t + ε t = β 0 + k i=1 β i t i i! + m i=1 A i sin ( ) 2πi s t + φ i + ε t, where the number of harmonics m should not goes beyond s/2, i.e. half the seasonality. Monthly, quarterly data; s/2 harmonics are usually not necessary This is illustrated in Figure, where we plot E(z t ) = 2 A i sin i=1 ( ) 2πi 12 t + φ i for A 1 = 1, φ 1 = 0, A 2 = 0.70, φ 2 =.6944π. 49

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51 Change of Time Origin in the Seasonal Trigonometric Model Examples: 12-point sinusoidal model (k = 0, s = 12, m = 1) z n+j = β 0 + β 11 sin 2πj 12 + β 21 cos 2πj 12 + ε n+j In this case: L = /2 1/2 0 1/2 3/2, f(0) = Linear trend model with two superimposed harmonics (k = 1, s = 12, m = 2): z n+j = β 0 +β 1 j+β 11 sin 2πj 12 +β 21 cos 2πj 12 +β 12 sin 4πj 12 +β 22 cos 4πj 12 +ε n+j 51

52 Target: Minimizing Locally Constant Seasonal Models z n+j = f (j)β + ε n+j S(β, n) = n 1 j=0 w j [z n j f ( j)β] 2 updating: ˆβ n+1 = L ˆβn + F 1 f(0)[z n+1 ẑ n (1)], (F = j 0 w j f ( j)f( j)) A collection of infinite sums needed to cacluate F for seasonal models is given in the following Table. 52

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54 It is usually suggested that the least squares estimate of β in the regression model z t = f (t)β + ε t be taken as initial vector ˆβ 0. To update the estimates, a smoothing constant must be determined. As Brown (1962) suggest that the value of w should lie between (.70) 1/g and (.95) 1/g If sufficient historical data are available, one can estimate w = 1 α by simulation and choose the smoothing constant that minimizes the sum of the squared one-step-ahead forecast errors SSE(α) = n [z t ẑ t 1 (1)] 2 t=1 After estimating the smoothing constant α, one should always check the adequacy of the model. The sample autocorrelation function of the onestep-ahead forecast errors should be calculated. Significant autocorrelations indicate that the particular forecast model is not appropriate. 54

55 Locally Constant Seasonal Models Using Seasonal Indicators z n+j = β 0 + β 1 j + 3 δ i IND ji + ε n+j = f (j)β i=1 where f(j) = [1 j IND j1 IND j2 IND j3 ], f(0) = [ ]. Then L = Hence the updating weights in ˆβ n+1 = L ˆβn + F 1 f(0)[z n+1 ẑ n (1)] can be calculated from f(0) and the symmetric matrix 55

56 F = 1 1 w w (1 w) 2 w 3 1 w 4 w 2 1 w 4 w(1+w) w 3 (3+w 4 ) w 2 (2+2w 4 ) (1 w) 3 (1 w 4 ) 2 w 1 w 4 w(1+3w 4 ) (1 w 4 ) 2 (1 w 4 ) 2 w 3 1 w symmetric w 2 1 w 4 0 w 1 w 4 Implications of α 1 In this situation F singular as w 0. But we have that lim w 0 F 1 f = f = ( 1, 1 s, 1 s, 2 s,, s 1 ) ; s ˆβ n+1 = L ˆβn + f [z n+1 ẑ n (1)]; ẑ n (1) = z n + z n+1 s z n s = z n+1 s + (z n z n s ) ẑ n (l) = ẑ n (l 1) + ẑ n (l s) ẑ n (l s 1) 56

57 Example: Car Sales Consider the monthly car sales in Quebec from January 1960 through December 1967 (n = 96 observations). The remaining 12 observations (1968) are used as a holdcut period to evaluate the forecast performance. An initial inspection of the series in Figure shows the data may be described by an additive model with a linear trend (k = 1) and a yearly seasonal pattern; the trend and the seasonal components appear fairly constant. z t = β 0 + β 1 t + 11 i=1 δ i IND ti + ε t. 57

58 Car sales (1000s) Year Figure: Monthly car sales in Quebec, Canada; January 1960 to December

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65 Example: New Plant and Equipment Expenditures Consider quarterly new plant and equipment expenditures for the first quarter of 1964 through the fourth quarter of 1974 (n = 44). The time series plot in Figure indicate s that the size of the seasonal swings increases with the level of the series; hence a logarithmic transformation must considered. The next Figure shows that this transformation has stabilized the variance. z t = ln y t = β 0 + β δ i IND ti + ε t i=1 65

66 40 New plant and equipment expenditures (billion \$) Year Figure: Quarterly new plant and equipment expenditure in U.S. industries (in billions of dollars), first quarter 1 to fourth quarter

67 Logarithm of new plant and equipment expenditures Year Logarithm of quarterly new plant and equipment expenditures, frist quarter 1964 to fourth quarter

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