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Econ 240 CEcon 240 C
Lecture 13Lecture 13
1
2
Re-VisitRe-Visit
Santa Barbara South Coast House Santa Barbara South Coast House PricePrice
UC Budget, General FundUC Budget, General FundExponential Smoothing, p.43-Exponential Smoothing, p.43------------------------------------------------------------------------- Intervention AnalysisIntervention Analysis
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Santa Barbara South Coast Median House Price, 1974-2008
Thousands of Nominal DollarsThousands of Nominal Dollars
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SB South Coast Median House Price, Log Scale
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Santa Barbara South Coast Median House Price, 2008$
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Santa Barbara South Coast LN Median House Price 2008 $
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Residual Actual Fitted
Actual, Fitted and residual from Regression, 1996-2005
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UC Budget, General Fund Vs. CA Personal Income in $B, 1968-69 Through 2009-10
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CA General Fund Expenditures Vs. Personal Income 1968-69 through 2009-10
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Relative Size of CA Government, 1968-69 through 2009-10
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GovernmentGovernment
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LNCAPY
Log of California Personal Income in Nominal Dollars, 1968-69 through 2009-10
CA Personal Income, Nominal CA Personal Income, Nominal $$
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OutlineOutline
Exponential SmoothingExponential SmoothingBack of the envelope formula: geometric Back of the envelope formula: geometric
distributed lag: L(t) = a*y(t-1) + (1-a)*L(t-1); distributed lag: L(t) = a*y(t-1) + (1-a)*L(t-1); F(t) = L(t)F(t) = L(t)
ARIMA (p,d,q) = (0,1,1); ARIMA (p,d,q) = (0,1,1); ∆y(t) = e(t) –(1-a)e(t-∆y(t) = e(t) –(1-a)e(t-1)1)
Error correction: L(t) =L(t-1) + a*e(t)Error correction: L(t) =L(t-1) + a*e(t) Intervention AnalysisIntervention Analysis
14
Part I: Exponential SmoothingPart I: Exponential Smoothing
Exponential smoothing is a technique Exponential smoothing is a technique that is useful for forecasting short that is useful for forecasting short time series where there may not be time series where there may not be enough observations to estimate a enough observations to estimate a Box-Jenkins modelBox-Jenkins model
Exponential smoothing can be Exponential smoothing can be understood from many perspectives; understood from many perspectives; one perspective is a formula that one perspective is a formula that could be calculated by handcould be calculated by hand
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Three Rates of Growth
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Santa Barbara South Coast House Price, 000 04 $
18Simple exponential Simple exponential smoothingsmoothing
Simple exponential smoothing, also Simple exponential smoothing, also known as single exponential smoothing, known as single exponential smoothing, is most appropriate for a time series is most appropriate for a time series that is a random walk with first order that is a random walk with first order moving average error structuremoving average error structure
The levels term, L(t), is a weighted The levels term, L(t), is a weighted average of the observation lagged one, average of the observation lagged one, y(t-1) plus the previous levels, L(t-1):y(t-1) plus the previous levels, L(t-1):
L(t) = a*y(t-1) + (1-a)*L(t-1)L(t) = a*y(t-1) + (1-a)*L(t-1)
19
Single exponential smoothingSingle exponential smoothing
The parameter a is chosen to minimize The parameter a is chosen to minimize the sum of squared errors where the the sum of squared errors where the error is the difference between the error is the difference between the observation and the levels term: e(t) = observation and the levels term: e(t) = y(t) – L(t)y(t) – L(t)
The forecast for period t+1 is given by The forecast for period t+1 is given by the formula: L(t+1) = a*y(t) + (1-a)*L(t)the formula: L(t+1) = a*y(t) + (1-a)*L(t)
Example from John Heinke and Arthur Example from John Heinke and Arthur Reitsch, Reitsch, Business Forecasting, 6Business Forecasting, 6thth Ed. Ed.
20observations Sales
1 500
2 350
3 250
4 400
5 450
6 350
7 200
8 300
9 350
10 200
11 150
12 400
13 550
14 350
15 250
16 550
17 550
18 400
19 350
20 600
21 750
22 500
23 400
24 650
21
Single exponential smoothingSingle exponential smoothing
For observation #1, set L(1) = For observation #1, set L(1) = Sales(1) = 500, as an initial conditionSales(1) = 500, as an initial condition
As a trial value use a = 0.1As a trial value use a = 0.1So L(2) = 0.1*Sales(1) + 0.9*Level(1) So L(2) = 0.1*Sales(1) + 0.9*Level(1)
L(2) = 0.1*500 + 0.9*500 = 500 L(2) = 0.1*500 + 0.9*500 = 500And L(3) = 0.1*Sales(2) + And L(3) = 0.1*Sales(2) +
0.9*Level(2) L(3) = 0.1*350 + 0.9*Level(2) L(3) = 0.1*350 + 0.9*500 = 4850.9*500 = 485
22observations Sales Level
1 500 500
2 350
3 250
4 400
5 450
6 350
7 200
8 300
9 350
10 200
11 150
12 400
13 550
14 350
15 250
16 550
17 550
18 400
19 350
20 600
21 750
22 500
23 400
24 650
23observations Sales Level
1 500 500
2 350 500
3 250 485
4 400
5 450
6 350
7 200
8 300
9 350
10 200
11 150
12 400
13 550
14 350
15 250
16 550
17 550
18 400
19 350
20 600
21 750
22 500
23 400
24 650
a = 0.1
24
Single exponential smoothingSingle exponential smoothing
So the formula can be used to So the formula can be used to calculate the rest of the levels calculate the rest of the levels values, observation #4-#24values, observation #4-#24
This can be set up on a spread-sheetThis can be set up on a spread-sheet
25observations Sales Level
1 500 500
2 350 500
3 250 485
4 400 461.5
5 450 455.4
6 350 454.8
7 200 444.3
8 300 419.9
9 350 407.9
10 200 402.1
11 150 381.9
12 400 358.7
13 550 362.8
14 350 381.6
15 250 378.4
16 550 365.6
17 550 384.0
18 400 400.6
19 350 400.5
20 600 395.5
21 750 415.9
22 500 449.3
23 400 454.4
24 650 449.0
a = 0.1
26
Single exponential smoothingSingle exponential smoothing
The forecast for observation #25 is: The forecast for observation #25 is: L(25) = 0.1*sales(24)+0.9*L(24)L(25) = 0.1*sales(24)+0.9*L(24)
Forecast(25)=Levels(25)=0.1*650+0.9Forecast(25)=Levels(25)=0.1*650+0.9*449*449
Forecast(25) = 469.1Forecast(25) = 469.1
Single Exponential Smoothing
0
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500
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Observation
Val
ue Sales
Levels
28Single exponential Single exponential distributiondistribution
The errors can now be calculated: The errors can now be calculated: e(t) = sales(t) – levels(t) e(t) = sales(t) – levels(t)
29observations Sales Level error
1 500 500 0
2 350 500 -150
3 250 485 -235
4 400 461.5 -61.5
5 450 455.4 -5.35
6 350 454.8 -104.8
7 200 444.3 -244.3
8 300 419.9 -119.9
9 350 407.9 -57.9
10 200 402.1 -202.1
11 150 381.9 -231.9
12 400 358.7 41.3
13 550 362.8 187.2
14 350 381.6 -31.6
15 250 378.4 -128.4
16 550 365.6 184.4
17 550 384.0 166.0
18 400 400.6 -0.6
19 350 400.5 -50.5
20 600 395.5 204.5
21 750 415.9 334.1
22 500 449.3 50.7
23 400 454.4 -54.4
24 650 449.0 201.0
a = 0.1
30observations Sales Level error
error squared
1 500 500 0 0
2 350 500 -150 22500
3 250 485 -235 55225
4 400 461.5 -61.5 3782.25
5 450 455.4 -5.35 28.62
6 350 454.8 -104.8 10986.18
7 200 444.3 -244.3 59698.86
8 300 419.9 -119.9 14376.05
9 350 407.9 -57.9 3353.58
10 200 402.1 -202.1 40852.14
11 150 381.9 -231.9 53780.95
12 400 358.7 41.3 1704.33
13 550 362.8 187.2 35027.05
14 350 381.6 -31.6 996.06
15 250 378.4 -128.4 16487.67
16 550 365.6 184.4 34016.68
17 550 384.0 166.0 27553.51
18 400 400.6 -0.6 0.37
19 350 400.5 -50.5 2554.91
20 600 395.5 204.5 41823.74
21 750 415.9 334.1 111594.53
22 500 449.3 50.7 2565.62
23 400 454.4 -54.4 2960.80
24 650 449.0 201.0 40412.28
a = 0.1
31observations Sales Level errorerror squared
1 500 500 0 0
2 350 500 -150 22500
3 250 485 -235 55225
4 400 461.5 -61.5 3782.25
5 450 455.4 -5.35 28.62
6 350 454.8 -104.8 10986.18
7 200 444.3 -244.3 59698.86
8 300 419.9 -119.9 14376.05
9 350 407.9 -57.9 3353.58
10 200 402.1 -202.1 40852.14
11 150 381.9 -231.9 53780.95
12 400 358.7 41.3 1704.33
13 550 362.8 187.2 35027.05
14 350 381.6 -31.6 996.06
15 250 378.4 -128.4 16487.67
16 550 365.6 184.4 34016.68
17 550 384.0 166.0 27553.51
18 400 400.6 -0.6 0.37
19 350 400.5 -50.5 2554.91
20 600 395.5 204.5 41823.74
21 750 415.9 334.1 111594.53
22 500 449.3 50.7 2565.62
23 400 454.4 -54.4 2960.80
24 650 449.0 201.0 40412.28
sum sq res 582281.2
a = 0.1
32
Single exponential smoothingSingle exponential smoothing
For a = 0.1, the sum of squared For a = 0.1, the sum of squared errors is: errors is: errors)errors)2 2 = 582,281.2= 582,281.2
A grid search can be conducted for A grid search can be conducted for the parameter value a, to find the the parameter value a, to find the value between 0 and 1 that value between 0 and 1 that minimizes the sum of squared errorsminimizes the sum of squared errors
The calculations of levels, L(t), and The calculations of levels, L(t), and errors, e(t) = sales(t) – L(t) for a =0.6errors, e(t) = sales(t) – L(t) for a =0.6
33observations Sales Levels
1 500 500
2 350 500
3 250 410
4 400 314
5 450 365.6
6 350 416.2
7 200 376.5
8 300 270.6
9 350 288.2
10 200 325.3
11 150 250.1
12 400 190.0
13 550 316.0
14 350 456.4
15 250 392.6
16 550 307.0
17 550 452.8
18 400 511.1
19 350 444.4
20 600 387.8
21 750 515.1
22 500 656.0
23 400 562.4
24 650 465.0
a = 0.6
34
Single exponential smoothingSingle exponential smoothing
Forecast(25) = Levels(25) = Forecast(25) = Levels(25) = 0.6*sales(24) + 0.4*levels(24) = 0.6*sales(24) + 0.4*levels(24) = 0.6*650 + 0.4*465 = 7760.6*650 + 0.4*465 = 776
35observations Sales Levels error
error square
1 500 500 0 0
2 350 500 -150 22500
3 250 410 -160 25600
4 400 314 86 7396
5 450 365.6 84.4 7123.36
6 350 416.2 -66.2 4387.74
7 200 376.5 -176.5 31150.84
8 300 270.6 29.4 864.45
9 350 288.2 61.8 3814.38
10 200 325.3 -125.3 15699.02
11 150 250.1 -100.1 10023.67
12 400 190.0 210.0 44080.13
13 550 316.0 234.0 54747.14
14 350 456.4 -106.4 11322.57
15 250 392.6 -142.6 20324.22
16 550 307.0 243.0 59036.75
17 550 452.8 97.2 9445.88
18 400 511.1 -111.1 12348.55
19 350 444.4 -94.4 8920.73
20 600 387.8 212.2 45037.39
21 750 515.1 234.9 55172.40
22 500 656.0 -156.0 24349.97
23 400 562.4 -162.4 26379.58
24 650 465.0 185.0 34237.15
Sum of Sq Res 533961.9
a = 0.6
36
Single exponential smoothingSingle exponential smoothing
Grid search plotGrid search plot
Grid Search for Smoothing Parameter
490000
500000
510000
520000
530000
540000
550000
560000
570000
580000
590000
0 0.2 0.4 0.6 0.8 1 1.2
Smoothing Parameter
Su
m o
f S
qu
ared
Res
idu
als
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Single Exponential SmoothingSingle Exponential Smoothing EVIEWS: Algorithmic search for the EVIEWS: Algorithmic search for the
smoothing parameter asmoothing parameter a In EVIEWS, select time series sales(t), and In EVIEWS, select time series sales(t), and
openopen In the sales window, go to the PROCS menu and In the sales window, go to the PROCS menu and
select exponential smoothingselect exponential smoothing Select singleSelect single the best parameter a = 0.26 with sum of squared the best parameter a = 0.26 with sum of squared
errors = 472982.1 and root mean square error = errors = 472982.1 and root mean square error = 140.4 = (472982.1/24)140.4 = (472982.1/24)1/21/2
The forecast, or end of period levels mean = 532.4The forecast, or end of period levels mean = 532.4
39
40
41Forecast = L(25) = 0.26*Sales(24) + 0.74L(24) = 532.4=0.26*650 + 0.74*491.07 =532.4
42
43Part II. Three Perspectives on Part II. Three Perspectives on Single Exponential SmoothingSingle Exponential SmoothingThe formula perspectiveThe formula perspective
L(t) = a*y(t-1) + (1 - a)*L(t-1)L(t) = a*y(t-1) + (1 - a)*L(t-1)e(t) = y(t) - L(t)e(t) = y(t) - L(t)
The Box-Jenkins PerspectiveThe Box-Jenkins PerspectiveThe Updating Forecasts PerspectiveThe Updating Forecasts Perspective
44
Box Jenkins PerspectiveBox Jenkins PerspectiveUse the error equation to substitute for L(t) Use the error equation to substitute for L(t)
in the formula, L(t) = a*y(t-1) + (1 - a)*L(t-in the formula, L(t) = a*y(t-1) + (1 - a)*L(t-1)1)L(t) = y(t) - e(t)L(t) = y(t) - e(t)y(t) - e(t) = a*y(t-1) + (1 - a)*[y(t-1) - e(t-1)] y(t) - e(t) = a*y(t-1) + (1 - a)*[y(t-1) - e(t-1)]
y(t) = e(t) + y(t-1) - (1-a)*e(t-1)y(t) = e(t) + y(t-1) - (1-a)*e(t-1)or or y(t) = y(t) - y(t-1) = e(t) - (1-a) e(t-1)y(t) = y(t) - y(t-1) = e(t) - (1-a) e(t-1)
So y(t) is a random walk plus MAONE noise, So y(t) is a random walk plus MAONE noise, i.e y(t) is a (0,1,1) process where (p,d,q) are i.e y(t) is a (0,1,1) process where (p,d,q) are the orders of AR, differencing, and MA.the orders of AR, differencing, and MA.
45
Box-Jenkins PerspectiveBox-Jenkins Perspective
In Lab Seven, we will apply simple In Lab Seven, we will apply simple exponential smoothing to retail sales, exponential smoothing to retail sales, and which can be modeled as (0,1,1).and which can be modeled as (0,1,1).
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130000
140000
150000
160000
170000
90 91 92 93 94 95 96
RETAIL RETAILSM
Retail Sales: Simple Exponential Smoothing
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Box-Jenkins PerspectiveBox-Jenkins Perspective
If the smoothing parameter approaches If the smoothing parameter approaches one, then y(t) is a random walk:one, then y(t) is a random walk:y(t) = y(t) - y(t-1) = e(t) - (1-a) e(t-1)y(t) = y(t) - y(t-1) = e(t) - (1-a) e(t-1)if a = 1, then if a = 1, then y(t) = y(t) - y(t-1) = e(t) y(t) = y(t) - y(t-1) = e(t)
In Lab Seven, we will use the price of In Lab Seven, we will use the price of gold to make this pointgold to make this point
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360
380
400
420
440
460
10 20 30 40 50 60 70
GOLD
Weekly Closing Price of Gold , Nov. 14, 2003-April 29, 2005
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54
55
56Box-Jenkins PerspectiveBox-Jenkins PerspectiveThe levels or forecast, L(t), is a The levels or forecast, L(t), is a
geometric distributed lag of past geometric distributed lag of past observations of the series, y(t), hence observations of the series, y(t), hence the name “exponential” smoothingthe name “exponential” smoothingL(t) = a*y(t-1) + (1 - a)*L(t-1)L(t) = a*y(t-1) + (1 - a)*L(t-1)L(t) = a*y(t-1) + (1 - a)*ZL(t)L(t) = a*y(t-1) + (1 - a)*ZL(t)L(t) - (1 - a)*ZL(t) = a*y(t-1) L(t) - (1 - a)*ZL(t) = a*y(t-1) [1 - (1-a)Z] L(t) = a*y(t-1) [1 - (1-a)Z] L(t) = a*y(t-1) L(t) = {1/ [1 - (1-a)Z]} a*y(t-1) L(t) = {1/ [1 - (1-a)Z]} a*y(t-1) L(t) = [1 +(1-a)Z + (1-a)L(t) = [1 +(1-a)Z + (1-a)2 2 ZZ2 2 + …] a*y(t-1)+ …] a*y(t-1)L(t) = a*y(t-1) + (1-a)*a*y(t-2) + (1-L(t) = a*y(t-1) + (1-a)*a*y(t-2) + (1-
a)a)22a*y(t-3) + ….a*y(t-3) + ….
57
58The Updating Forecasts The Updating Forecasts PerspectivePerspective
Use the error equation to substitute for Use the error equation to substitute for y(t) in the formula, L(t) = a*y(t-1) + (1 - y(t) in the formula, L(t) = a*y(t-1) + (1 - a)*L(t-1)a)*L(t-1)y(t) = L(t) + e(t)y(t) = L(t) + e(t)L(t) = a*[L(t-1) + e(t-1)] + (1 - a)*L(t-1)L(t) = a*[L(t-1) + e(t-1)] + (1 - a)*L(t-1)
So L(t) = L(t-1) + a*e(t-1), So L(t) = L(t-1) + a*e(t-1), i.e. the forecast for period t is equal to the i.e. the forecast for period t is equal to the
forecast for period t-1 plus a fraction a of the forecast for period t-1 plus a fraction a of the forecast error from period t-1.forecast error from period t-1.
59Part III. Double Exponential Part III. Double Exponential
Smoothing Smoothing With double exponential smoothing, one With double exponential smoothing, one
estimates a “trend” term, R(t), as well estimates a “trend” term, R(t), as well as a levels term, L(t), so it is possible to as a levels term, L(t), so it is possible to forecast, f(t), out more than one periodforecast, f(t), out more than one period
f(t+k) = L(t) + k*R(t), k>=1f(t+k) = L(t) + k*R(t), k>=1L(t) = a*y(t) + (1-a)*[L(t-1) + R(t-1)]L(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)R(t) = b*[L(t) - L(t-1)] + (1-b)*R(t-1)
so the trend, R(t), is a geometric distributed so the trend, R(t), is a geometric distributed lag of the change in levels, lag of the change in levels, L(t)L(t)
1k 1k 1k
1k
60
If the smoothing parameters a = b, If the smoothing parameters a = b, then we have double exponential then we have double exponential smoothingsmoothing
If the smoothing parameters are If the smoothing parameters are different, then it is the simplest different, then it is the simplest version of Holt-Winters smoothingversion of Holt-Winters smoothing
Part III. Double Exponential Part III. Double Exponential SmoothingSmoothing
61Part III. Double Exponential Part III. Double Exponential SmoothingSmoothing
Holt- Winters can also be used to forecast Holt- Winters can also be used to forecast seasonal time series, e.g. monthlyseasonal time series, e.g. monthly
f(t+k) = L(t) + k*R(t) + S(t+k-12) k>=1f(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-L(t) = a*[y(t)-S(t-12)]+ (1-a)*[L(t-1) + R(t-
1)]1)]R(t) = b*[L(t) - L(t-1)] + (1-b)*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)S(t) = c*[y(t) - L(t)] + (1-c)*S(t-12)
62
Part V. Intervention AnalysisPart V. Intervention Analysis
63
Intervention AnalysisIntervention Analysis
The approach to intervention The approach to intervention analysis parallels Box-Jenkins in that analysis parallels Box-Jenkins in that the actual estimation is conducted the actual estimation is conducted after pre-whitening, to the extent after pre-whitening, to the extent that non-stationarity such as trend that non-stationarity such as trend and seasonality are removedand seasonality are removed
Example: preview of Lab 7Example: preview of Lab 7
64Telephone Directory Telephone Directory AssistanceAssistance
A telephone company was receiving A telephone company was receiving increased demand for free directory increased demand for free directory assistance, i.e. subscribers asking assistance, i.e. subscribers asking operators to look up numbers. This operators to look up numbers. This was increasing costs and the was increasing costs and the company changed policy, providing a company changed policy, providing a number of free assisted calls to number of free assisted calls to subscribers per month, but charging subscribers per month, but charging a price per call after that number. a price per call after that number.
65Telephone Directory Telephone Directory AssistanceAssistance
This policy change occurred at a This policy change occurred at a known time, March 1974known time, March 1974
The time series is for calls with The time series is for calls with directory assistance per monthdirectory assistance per month
Did the policy change make a Did the policy change make a difference?difference?
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67
The simple-minded approachThe simple-minded approach
=549 - 162
=387
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PrinciplePrinciple
The event may cause a change, and The event may cause a change, and affect time series characteristicsaffect time series characteristics
Consequently, consider the pre-event Consequently, consider the pre-event period, January 1962 through period, January 1962 through February 1974, the event March February 1974, the event March 1974, and the post-event period, 1974, and the post-event period, April 1974 through December 1976April 1974 through December 1976
First difference and then seasonally First difference and then seasonally difference the entire seriesdifference the entire series
72Analysis: Entire Differenced Analysis: Entire Differenced SeriesSeries
73
74
75
76
77Analysis: Pre-Event Analysis: Pre-Event DifferencesDifferences
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So Seasonal NonstationaritySo Seasonal Nonstationarity
It was masked in the entire sample It was masked in the entire sample by the variance caused by the by the variance caused by the difference from the eventdifference from the event
The seasonality was revealed in the The seasonality was revealed in the pre-event differenced series pre-event differenced series
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Pre-Event AnalysisPre-Event Analysis
Seasonally differenced, differenced Seasonally differenced, differenced series series
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Pre-Event Box-Jenkins ModelPre-Event Box-Jenkins Model
[1-Z[1-Z12 12 ][1 –Z]Assist(t) = WN(t) – a*WN(t-][1 –Z]Assist(t) = WN(t) – a*WN(t-12)12)
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Modeling the EventModeling the Event
Step functionStep function
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Entire SeriesEntire Series
Assist and StepAssist and StepDassist and DstepDassist and DstepSddast sddstepSddast sddstep
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Model of Series and EventModel of Series and Event
Pre-Event Model: [1-ZPre-Event Model: [1-Z12 12 ][1 –Z]Assist(t) = ][1 –Z]Assist(t) = WN(t) – a*WN(t-12)WN(t) – a*WN(t-12)
In Levels Plus Event: Assist(t)=[WN(t) – In Levels Plus Event: Assist(t)=[WN(t) – a*WN(t-12)]/[1-Z]*[1-Za*WN(t-12)]/[1-Z]*[1-Z1212] + (-b)*step] + (-b)*step
Estimate: [1-ZEstimate: [1-Z12 12 ][1 –Z]Assist(t) = WN(t) ][1 –Z]Assist(t) = WN(t) – a*WN(t-12) + (-b)* [1-Z– a*WN(t-12) + (-b)* [1-Z12 12 ][1 –Z]*step][1 –Z]*step
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Policy Change EffectPolicy Change Effect
Simple: decrease of 387 (thousand) Simple: decrease of 387 (thousand) calls per monthcalls per month
Intervention model: decrease of 397 Intervention model: decrease of 397 with a standard error of 22with a standard error of 22
101
102Stochastic Trends: Random Stochastic Trends: Random
Walks with DriftWalks with DriftWe have discussed earlier in the We have discussed earlier in the
course how to model the Total Return course how to model the Total Return to the Standard and Poor’s 500 Indexto the Standard and Poor’s 500 Index
One possibility is this time series could One possibility is this time series could be a random walk around a be a random walk around a deterministic trend”deterministic trend”
Sp500(t) = exp{a + d*t +WN(t)/[1-Z]}Sp500(t) = exp{a + d*t +WN(t)/[1-Z]}And taking logarithms,And taking logarithms,
103Stochastic Trends: Random Stochastic Trends: Random
Walks with DriftWalks with Drift Lnsp500(t) = a + d*t + WN(t)/[1-Z]Lnsp500(t) = a + d*t + WN(t)/[1-Z] Lnsp500(t) –a –d*t = WN(t)/[1-Z]Lnsp500(t) –a –d*t = WN(t)/[1-Z] Multiplying through by the difference Multiplying through by the difference
operator, operator, = [1-Z]= [1-Z] [1-Z][Lnsp500(t) –a –d*t] = WN(t-1)[1-Z][Lnsp500(t) –a –d*t] = WN(t-1)
[LnSp500(t) – a –d*t] - [LnSp500(t-1) – a –[LnSp500(t) – a –d*t] - [LnSp500(t-1) – a –d*(t-1)] = WN(t)d*(t-1)] = WN(t)
Lnsp500(t) = d + WN(t)Lnsp500(t) = d + WN(t)
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So the fractional change in the total return So the fractional change in the total return to the S&P 500 is drift, d, plus white noiseto the S&P 500 is drift, d, plus white noise
More generally, More generally, y(t) = a + d*t + {1/[1-Z]}*WN(t)y(t) = a + d*t + {1/[1-Z]}*WN(t)[y(t) –a –d*t] = {1/[1-Z]}*WN(t)[y(t) –a –d*t] = {1/[1-Z]}*WN(t)[y(t) –a –d*t]- [y(t-1) –a –d*(t-1)] = WN(t)[y(t) –a –d*t]- [y(t-1) –a –d*(t-1)] = WN(t)[y(t) –a –d*t]= [y(t-1) –a –d*(t-1)] + WN(t)[y(t) –a –d*t]= [y(t-1) –a –d*(t-1)] + WN(t)Versus the possibility of an ARONE:Versus the possibility of an ARONE:
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[y(t) –a –d*t]=b*[y(t-1)–a–d*(t-1)]+WN(t)[y(t) –a –d*t]=b*[y(t-1)–a–d*(t-1)]+WN(t) Y(t) = a + d*t + b*[y(t-1)–a–d*(t-1)]+WN(t)Y(t) = a + d*t + b*[y(t-1)–a–d*(t-1)]+WN(t) Or y(t) = [a*(1-b)+b*d]+[d*(1-b)]*t+b*y(t-1) Or y(t) = [a*(1-b)+b*d]+[d*(1-b)]*t+b*y(t-1)
+wn(t)+wn(t) Subtracting y(t-1) from both sides’Subtracting y(t-1) from both sides’ y(t) = [a*(1-b)+b*d] + [d*(1-b)]*t + (b-1)*y(t-1) y(t) = [a*(1-b)+b*d] + [d*(1-b)]*t + (b-1)*y(t-1)
+wn(t)+wn(t) So the coefficient on y(t-1) is once again So the coefficient on y(t-1) is once again
interpreted as b-1, and we can test the null that interpreted as b-1, and we can test the null that this is zero against the alternative it is this is zero against the alternative it is significantly negative. Note that we specify the significantly negative. Note that we specify the equation with both a constant, equation with both a constant,
[a*(1-b)+b*d] and a trend [d*(1-b)]*t [a*(1-b)+b*d] and a trend [d*(1-b)]*t
106Part IV. Dickey Fuller Tests: Part IV. Dickey Fuller Tests: TrendTrend
107
ExampleExample
Lnsp500(t) from Lab 2Lnsp500(t) from Lab 2
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