Econ 240 C

1
Econ 240 C

• Lecture 6

2
Outline

• I. Transcendental ideas
• II. Lab Two
• III. Analysis and synthesis
• IV. Characterizing time series behavior
• V. Analysis in the Lab Process

3
Ideas to remember

• Models are just a guide to thinking, not an end
  in themselves
• If the model is the box, think outside of the
  box compare competing models
• Sell complexity, believe simplicity

4
Outline Lab Two

• Linear Trend model for sp500
• Exponential trend model lnsp500
• Other models
• Exponential trend plus random walk
• Naïve models

5
Linear trend model

• Process combine what you learned in 240A, B, C
• Spreadsheet
• Plot or trace not linear
• Histogram not normal
• Correlogram not orthogonal
• Unit root test not stationary
• Sp500(t) a b t e(t)
• Goodness of fit
• Violation of regression assumptions
• One period ahead forecast from the trend model

6
(No Transcript)
7
(No Transcript)
8
(No Transcript)
9
(No Transcript)
10
(No Transcript)
11
(No Transcript)
12
Linear trend model

• Process combine what you learned in 240A, B, C
• Spreadsheet
• Plot or trace not linear
• Histogram not normal
• Correlogram not orthogonal
• Unit root test not stationary
• Sp500(t) a b t e(t)
• Goodness of fit
• Violation of regression assumptions
• One period ahead forecast from the trend model

13
Diagnostics R2 0.72, low for time
series D-W0.016, very low
14
(No Transcript)
15
Residuals are not orthogonal
16
Residuals are not normal
17
(No Transcript)
18
Diagnostics R2 0.72, low for time
series D-W0.016, very low
19
One period ahead forecast

• E2003.02 Sp500(2003.03) -772.7503
  9.170486398
• E2003.02 Sp500(2003.03) -772.7503 3649.8534
• E2003.02 Sp500(2003.03) 2877.103
• Approximate 95 confidence interval /- 2ser
• Upper bound 2877.1 2651.4 4179.9
• Lower bound 2877.1 2651.4 1574.3

20
(No Transcript)
21
E2003.02 sp500(2003.03) Fitted(2003.02)
slope 2867.93 9.17 2877.1
E
22
(No Transcript)
23
(No Transcript)
24
(No Transcript)
25
Exponential (log-linear) trend model

• Process combine what you learned in 240A, B, C
• Spreadsheet
• Plot or trace more log-linear
• Histogram not normal
• Correlogram not orthogonal
• Unit root test not stationary
• Lnsp500(t) a b t e(t)
• Goodness of fit
• Violation of regression assumptions
• One period ahead forecast from the trend model

26
(No Transcript)
27
(No Transcript)
28
(No Transcript)
29
(No Transcript)
30
Exponential (log-linear) trend model

• Process combine what you learned in 240A, B, C
• Spreadsheet
• Plot or trace more log-linear
• Histogram not normal
• Correlogram not orthogonal
• Unit root test not stationary
• Lnsp500(t) a b t e(t)
• Goodness of fit
• Violation of regression assumptions
• One period ahead forecast from the trend model

31
R2 0.97, better fit D-W 0.042, very low
32
(No Transcript)
33
Residuals are not orthogonal
34
Residuals are not normal
35
Dependent Variable LNSP500 Method Least
Squares Sample(adjusted) 197001
200302 Included observations 398 after
adjusting endpoints Variable Coefficient Std
. Error t-Statistic Prob. C 4.049837 0.02
2383 180.9370 0.0000 TIME 0.010867 9.76E-05 111.3
580 0.0000 R-squared 0.969054 Mean
dependent var 6.207030 Adjusted
R-squared 0.968976 S.D. dependent
var 1.269965 S.E. of regression 0.22368
Akaike info criterion -0.152131 Sum squared
resid 19.8141 Schwarz criterion -0.132099 Log
likelihood 32.27406 F-statistic 12400.61 D
urbin-Watson stat 0.041769 Prob(F-statistic) 0.
000000
36
Fitted model

• Lnsp500(t) c bt
• 2003.03 is t398
• Forecast lnsp500(t398) 4.049837
  3980.010867
• Forecastlnsp500(2003.03) 8.374903
• Upper bound of 95 confidence interval
  8.374903 2 SER 8.374903 20.22368
• Upper bound 8.822263

37
Fitted Model

• Lower bound forecast -2ser 8.374903 -0.44736
  7.927543

38
(No Transcript)
39
Fitted model

• Forecast of sp500(2003.03) exp(8.374903)
  4336.848
• Upper bound exp(8.822263) 6783.599
• Lower bound exp(7.927543) 2772.606

40
(No Transcript)
41
Alternative models

• Sp500(t) expa bt rw(t)
• Lnsp500(t) a bt rw(t)
• ?lnsp500(t) b ? rw(t)
• ?lnsp500(t) b wn(t)

42
0.00862512 0.1035 annual rate of growth
43
Residuals are orthogonal
44
(No Transcript)
45
Mean of dlnsp500, H0 ? 0, 0.008625 0/std
dev/n1/2 Students t-statistic 3.76
46
Alternative models

• Naïve forecast best forecast of next period is
  this periods value
• True for a random walk
• Lnsp500(387) 7.86996118907
• Exp lnsp500(387) 2617.464

47
(No Transcript)
48
Conclusion

• Best model lnsp500(t) a bt rw(t)
• Best long run forecast is trend a bt
• Best short run forecast is random walk rw(t)

49
III Box-Jenkins Magic

• ARMA models of time series all built from one
  source, white noise

50
Analysis and Synthesis

• White noise, WN(t)
• Is a sequence of draws from a normal
  distribution, N(0, s2 ), indexed by time

51
Analysis

• Random walk, RW(t)
• Analysis formulation
• RW(t) RW(t-1) WN(t)
• RW(t) - RW(t-1) WN(t)
• RW(t) ZRW(t) WN(t)
• 1 ZRW(t) WN(t)
• DRW(t) WN(t) shows how you turn a random walk
  into white noise

52
Synthesis

• Random Walk, Synthesis formulation
• RW(t) 1/1 ZWN(t)
• RW(t) 1 Z Z2 .WN(t)
• RW(t) WN(t) ZWN(t) .
• RW(t) WN(t) WN(t-1) WN(t-2) shows how
  you build a random walk from white noise

53
Analysis

• Autoregressive process of the first order,
  analysis formulation
• ARONE(t) bARONE(t-1) WN(t)
• ARONE(t) - bARONE(t-1) WN(t)
• ARONE(t) - bZARONE(t) WN(t)
• 1 bZARONE(t) WN(t) is a quasi-difference
  and shows how you turn an autoregressive process
  of the first order into white noise

54
Synthesis

• Autoregressive process of the first order,
  synthetic formulation
• ARONE(t) 1/1 bZWN(t)
• ARONE(t) 1 bZ b2Z2 .WN(t)
• ARONE(t) WN(t)bZWN(t)b2Z2 WN(t) ..
• ARONE(t) WN(t) bWN(t-1) b2WN(t-2) .
  Shows how you turn white noise into an
  autoregressive process of the first order

55
Part IV Characterizing Time Series Behavior

• Mean function, m(t) E time_series(t)
• White noise m(t) E WN(t) 0, all t
• Random walk m(t) EWN(t)WN(t-1) .. equals
  0, all t
• First order autoregressive process,
  m(t) EWN(t) bWN(t-1) b2WN(t-2)
  equals 0, all t
• Note that for all three types of time series we
  calculate the mean function from the synthetic
  expression for the time series.
  

56
Characterization the AutocovarianceFunction

• EWN(t)WN(t-u) 0 for ugt0 , uses the
  orthogonality (independence) property of white
  noise
• ERW(t)RW(t-u) EWN(t)WN(t-1) WN(t-2)
  WN(t-u)WN(t-u-1) s2 s2 s2 ....
  , uses the orthogonality property for
  white noise plus the theoretically infinite
  history of a random walk

57
The Autocovariance Function

• EARONE(t)ARONE(t-u) bEARONE(t-1)
  ARONE(t-u) EWN(t)ARONE(t-u)
• gAR,AR(u) b gAR,AR(u-1) 0 ugt0, uses both
  the analytic and the synthetic formulations for
  ARONE(t). The analytic formulation is used to
  multiply by ARONE(t-u) and take expectations. The
  synthetic formulation is used to lag and show
  ARONE(t-1) depends only on WN(t-1) and earlier
  shocks.

58
The Autocorrelation Function

• rx,x(u) gAR,AR(u)/ gAR,AR(0)
• White Noise rWN,WN(u) 0u
• Random Walk rRW,RW(u) 1, all u
• Autoregressive of the first order rx,x(u) bu

59
Visual Preview of the Autocorrelation Function
60
Visual Preview of the Autocorrelation Function
61
Visual Preview of the Autocorrelation Function
62
Drop Lag Zero The Mirror Image of the Mark of
Zorro
Random Walk
1
First Order Autoregressive
White Noise
0
Lag
63
Part V .Analysis in the Lab Process

• Identification
• Estimation
• Verification
• Forecasting

64
Analysis in the Lab Process

• Identification
• Is the time series stationary?
• Trace
• Histogram
• Autocorrelation Function
• If it is, proceed
• If it is not, difference (prewhitening)

65
(No Transcript)
66
(No Transcript)
67
(No Transcript)
68
(No Transcript)
69
Process Analysis in the LAB

• Identification
• conclude evolutionary
• Fix-up pre-whiten by first differencing

70
(No Transcript)
71
(No Transcript)
72
(No Transcript)
73
(No Transcript)
74
(No Transcript)
75
Identification

• Conclude it is stationary
• Conjecture ARONE Model

76
Process Analysis in the LAB

• Estimation
• EVIEWS model
• time series(t) constant residual(t)
• residual(t) bresidual(t-1) WN(t)?
• Combine the two
• time series(t) - c btime series(t-1) - c
  WN(t)?
• EVIEWS Specification
• dinvsratio c ar(1)

77
(No Transcript)
78
(No Transcript)
79
Estimation

• Goodness of Fit
• Structure in the residuals? Are they orthogonal?
• Are the residuals normally distributed?

80
(No Transcript)
81
(No Transcript)
82
(No Transcript)
83
Process Analysis in the LAB

• Identification
• Estimation
• Verification
• Is there any structure left in the residuals? If
  not, we are back to our building block,
  orthogonal residuals, and we accept the model.

84
Process Analysis in the LAB

• Identification
• Estimation
• Verification
• Is there any structure left in the residuals? If
  not, we are back to our building block,
  orthogonal residuals, and we accept the model.
• Forecasting
• one period ahead forecasts

85
Process Analysis in the LAB

• Forecasting
• The estimated model
• dinvsratio(t) (-0.001430)
  -0.244617dinvsratio(t-1) 0.001430 WN(t)
  where WN(t) is approximately an independent error
  series and is normally distributed
• The forecast is based on the estimated model
• dinvsratio(2007.02) 0.001430
  -0.244617dinvsratio(2007.01) 0.001430
  WN(2007.02)

86
Process Analysis in the LAB

• Estimation
• EVIEWS model
• time series(t) constant residual(t)
• residual(t) bresidual(t-1) WN(t)?
• Combine the two
• time series(t) - c btime series(t-1) - c
  WN(t)?
• EVIEWS Specification
• dinvsratio c ar(1)

87
The Forecast

• Take expectations of the model, as of 2007.1
• E2007.01 dinvsratio(2007.02) 0.001430
  -0.244617E2007.01 dinvsratio(2007.01)
  0.001430 E2007.01 WN(2007.02)
• E2007.01 dinvsratio(2007.02) is the forecast
  conditional on what we know as of 2007.01
• dinvsratio(2007.01) 0.02, the value of the
  series in 2007.01
• E2007.01 WN(2007.02) 0, the best guess for the
  shock

88
The Forecast

• Calculate the forecast by hand
• for a one period ahead forecast, the standard
  error of the regression can be used for the
  standard error of the forecast
• calculate the upper band forecast 2SER
• calculate the lower band forecast - 2SER

89
The Forecast

• Use EVIEWS as a check

90
(No Transcript)
91
(No Transcript)
92
(No Transcript)