ECON 343 Lecture 4 : Smoothing and Extrapolation of Time Series. Jad Chaaban Spring
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1 ECON 343 Lecture 4 : Smoothing and Extrapolation of Time Series Jad Chaaban Spring
2 Outline Lecture 4 1. Simple extrapolation models 2. Moving-average models 3. Single Exponential smoothing 4. Double Exponential smoothing 5. Seasonal adjustment 6. Structural breaks
3 1. Simple extrapolation models Assumptions: Forecast the time series on the basis of its past behavior Deterministic nature: no randomness used in the models here Regress the series against a function of time and/or itself lagged Types of models: Linear trend model Quadratic trend model Exponential model Autoregressive trend model
4 Linear trend model Assumptions: The series will increase in constant absolute amounts each period Trend line depends only on time t Formula: Y t = α + βt t is usually chosen to equal 0 in the base period and to increase by 1 during each successive period Simple OLS estimation
5 Excel implementation IMF's International Price of Beef Index monthly, US cents per pound Jan-80 Jan-82 Jan-84 Jan-86 Jan-88 Jan-90 Jan-92 Jan-94 Jan-96 Jan-98 Jan-00 Jan-02 Jan-04
6 Excel implementation
7 Excel implementation
8 Excel implementation
9 Excel implementation IMF's International Price of Beef Index monthly, y = x R 2 = US cents per pound Jan-80 Jan-82 Jan-84 Jan-86 Jan-88 Jan-90 Jan-92 Jan-94 Jan-96 Jan-98 Jan-00 Jan-02 Jan-04
10 Quadratic trend model Assumptions: Extends the simple linear trend model to potential non-linearity Add a term t 2 Formula: Y t = α + βt + γt If β and γ are >0, then y t will always increase If β <0 and γ>0, y t will first decrease then increase If β and γ are <0, then y t will always decrease 2
11 Excel implementation
12 Excel implementation IMF's International Price of Beef Index monthly, y = x x R 2 = US cents per pound Jan-80 Jan-82 Jan-84 Jan-86 Jan-88 Jan-90 Jan-92 Jan-94 Jan-96 Jan-98 Jan-00 Jan-02 Jan-04
13 Excel implementation Polynomial of 6 time variables y = -1E-11x 6 + 7E-08x x x x x - 2E+07 R 2 = US cents per pound Jan-80 Jan-82 Jan-84 Jan-86 Jan-88 Jan-90 Jan-92 Jan-94 Jan-96 Jan-98 Jan-00 Jan-02 Jan-04
14 Exponential model Assumptions: The series will grow with constant percentage increases Use of the exponential function Formula: Y = The parameters A and r can be estimated by taking the logarithms of both sides of the formula Then we fit the through OLS the log-linear function: where c 1 =log(a) and c 2 =r t Ae rt log( Y ) = c + c t t 1 2
15 Excel implementation
16 Excel implementation IMF's International Price of Beef Index monthly, y = e x 60 R 2 = US cents per pound Jan-80 Jan-82 Jan-84 Jan-86 Jan-88 Jan-90 Jan-92 Jan-94 Jan-96 Jan-98 Jan-00 Jan-02 Jan-04
17 Autoregressive trend model Assumptions: The series will grow according to its lagged values only Formula: Y t = α + βyt 1 Variation of the model: logarithmic autoregressive trend model: log( Y ) = α + β log( Y ) t t 1 If α=0 then the value of β is the compounded rate of growth of the series Y
18 EViews implementation
19 EViews implementation
20 EViews implementation renamed normal autoregressive
21 EViews implementation
22 Logarithmic autoregressive EViews implementation
23 Logarithmic autoregressive EViews implementation
24 2. Moving Average models Assumptions: The likely value for our series next month is a simple average of its values over the past 12 months No regression used in the model Moving Average An n-period moving average is the average value over the previous n time periods. As you move forward in time, the oldest time period is dropped from the analysis. Weighted Moving Average An n-period weighted moving average allows you to place more weight on more recent time periods by weighting those time periods more heavily.
25 Moving Average models Formulas: Moving Average MA = with n time periods Weighted Moving Average WMA = n n i ( y,..., 1 t t m n i y t n ( α y,..., 1 α y ) α M ) t n
26 Excel implementation
27 Excel implementation IMF's International Price of Beef Index monthly, Pbeef 6 months MA US cents per pound Jan-80 Jan-83 Jan-86 Jan-89 Jan-92 Jan-95 Jan-98 Jan-01 Jan-04
28 3. Single Exponential Smoothing This single exponential smoothing method is appropriate for series that move randomly above and below a constant mean with no trend nor seasonal patterns. The smoothed series of is computed recursively, by evaluating: where is the damping (or smoothing) factor. The smaller is the, the smoother is the series. By repeated substitution, we can rewrite the recursion as
29 Single Exponential Smoothing Exponential smoothing-the forecast of y t is a weighted average of the past values of y t, where the weights decline exponentially with time. The forecasts from single smoothing are constant for all future observations. This constant is given by: Where T is the end of the estimation sample. EViews uses the mean of the initial observations of y t to start the recursion. Values of α around 0.01 to 0.30 work quite well. You can also let EViews estimate to minimize the sum of squares of one-step forecast errors
30 4. Double Exponential Smoothing This method applies the single smoothing method twice (using the same parameter) It is appropriate for series with a linear trend Double smoothing of a series y is defined by the recursions: where S is the single smoothed series and D is the double smoothed series. Note that double smoothing is a single parameter smoothing method with damping factor
31 Double Exponential Smoothing Forecasts from double smoothing are computed as: This expression shows that forecasts from double smoothing lie on a linear trend with intercept and slope
32 Eviews Exponential Smoothing
33 Eviews Exponential Smoothing
34 5. Seasonal adjustment If yt is a time series, then its generic element can be expressed as yt = Lt xct x St x εt where Lt : the global trend, Ct : a secular cycle (long term cyclical component), St : the seasonal variation, εt : an irregular component. The objective is to eliminate S First, estimate LxC by deriving an average. For monthly data, a 12-month average: ~ 1 yt = ( yt yt + yt yt 5 ) 12 which is free of irregular and seasonal fluctuations
35 Seasonal adjustment Then divide the original data by this estimate: L C S ε y t = S ε = = z t L C ~ y t The next step is to eliminate εt Average the values of St x εt corresponding to the same month For monthly data, suppose there are 48 months: ~ 1 z 1 = ( z 1 + z 13 + z 25 + z 37 ) 4 ~ 1 z 2 = ( z 2 + z 14 + z 26 + z 38 ) ~ z 1 = 1 ( z 12 + z 24 + z 36 + z 4 48 )
36 5. Seasonal adjustment The sum of these seasonal indices should be 12 in this case If not, adjust by a factor The final seasonal indices are used to deflate the data as follows: y y... y y a 1 a 2 a 13 a 14 etc =. = = = y y y y / / z / / z 1 2 z z 1 2
37 Eviews seasonal adjustment
38 Eviews seasonal adjustment
39 Importance of historical accidents IMF's International Price of Beef Index monthly, Mad Cow disease, UK US cents per pound Jan-80 Jan-82 Jan-84 Jan-86 Jan-88 Jan-90 Jan-92 Jan-94 Jan-96 Jan-98 Jan-00 Jan-02 Jan-04
40 Incorporating structural breaks Mad Cow timeline: Early 1990s: The British government insists the disease poses no threat to humans 1993: 120,000 cattle have been diagnosed with Bovine Spongiform Encephalopathy BSE in Britain May 1995: Stephen Churchill, 19, becomes the first victim of a new version of Creutzfeldt-Jakob Disease (vcjd). His is one of three vcjd deaths in 1995 April 2002: First confirmed case of vcjd appears in the US, in a 22-year-old British woman living in Florida Structural break: end of 1994
41 Dummy variable before after
42 Eviews estimation cons t t 2
43 Eviews estimation
44 Eviews estimation graph
45 Comparison Jan-80 Jan-82 Jan-84 Jan-86 Jan-88 Jan-90 Jan-92 Jan-94 Jan-96 Jan-98 Jan-00 Jan-02 Jan-04 Jan-06 Jan-08 Jan-10 Forecast quadratic, with break Forecast quadratic, no break
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