1. Econ 240 C
Lecture 4

5. Outline Part I: Time Averages Part II: Autocovariance Function
Part III: Random Walk
Part IV: Deciding Between a Random Walk Or an ARONE?

6. Practicum: Lab Two
Natural Logarithm of the Rotterdam Import Price for Dark Northern Spring Wheat
Trace
Histogram
Autocorrelation function

10. Practicum: Lab Two
First difference of Natural Logarithm of the Rotterdam Import price for Dark Northern Spring Wheat
Trace
Histogram
Autocorrelation function

14. Part I
Stationarity and Time Averages

15. Many Observations of White Noise
Using simulation, we can create many different white noise time series of arbitrary length, say 1000
Using these time series we could average them at each point in time to obtain an average of the group or ensemble at that time, creating the mean function, m(t) = E wn(t).

16. Time Averages
In economics, as a practical matter, we usually only have one observation of a time series
In this case it is necessary to be able to average over time
If the time series is stationary, then averaging over time makes sense

17. Five Simulated White Noise Time Series
WN WN2 WN3 WN4 WN5

18. Trace of Five Simulated White Noise Time Series

19. Ensemble Average

21. Appeal to Central Limit Theorem
If I had generated 100 or 1000 simulated white noise time series, then the ensemble average would come close to zero for every time period.
Mean Function, m(t) =E X(t)

22. Ensemble Average

23. Ensemble Averages Are a Luxury Good in Economics
Need to consider time average
Think of the time average for the first white noise time series

24. Time Average

25. Trace of Time Average

26. Mean Function For a Time Average
m = E WN(t) = c = 0

27. Time Average Will Not Work Well for an Evolutionary Time Series

28. Mean Function For An Evolutionary Time Series
M(t) = E x(t) is not independent of time

29. Part II Autocovariance Function E{ [x(t) - Ex(t)][x(t-u) - Ex(t)]} =
= gx,x(t,u)
If a time series is covariance stationary, then gx,x(t,u) = gx,x(u) i.e. depends only on lag.
Only relative time not absolute time counts.

30. Autocovariance of White Noise At Lag Zero
Autocovariance Function
E{ [wn(t) - Ewn(t)][wn(t-0) - Ewn(t)]}
E{ [wn(t) - m(t)][wn(t) - m(t)]}
E{ [wn(t) - 0][wn(t) - 0]}
E{wn(t)*wn(t)}

31. Autocovariance of White Noise At Lag Zero

32. Variance For Entire Series

33. Autocovariance of White Noise at Lag 1 = ?
Autocovariance Function
E{ wn(t)*wn(t-1) } =
gx,x(u=1) = ?

35. Average Cross Product For WN Series for 999

36. Autocovariance of White Noise at Lag 1 = ?
gx,x(u=1) = 0
Since white noise is independent as well as identically distributed

37. Autocovariance of White Noise at Lag 2 = ?
gx,x(u=2) = 0
Since white noise is independent as well as identically distributed

38. Theoretical Autocovariance, WN

39. Autocovariance, WN Theoretical Vs. Simulated

40. Autocorrelation Function
The Autocorrelation function is just a standardized autocovariance function, i.e. just the autocovariance function divided by the variance
rx,x (u) = gx,x(u) / gx,x(0)

41. Simulated Autocorrelation Function, WN
WN Simulated
Sample:
Included observations: 1000
Autocorrelation Partial Correlation AC PAC Q-Stat Prob
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42. Part III Random Walk as an evolutionary process
RW(t) = RW(t-1) + WN(t)
lag by one RW(t-1) = RW(t-2) +WN(t-1)
Substitute for RW(t-1)
RW(t) = RW(t-2) + WN(t) + WN(t-1)

43. Random Walk
Lag again to obtain RW(t-2) = RW(t-3) + WN(t-2) and Substitute for RW(t-2)
RW(t) = RW(t-3) + WN(t) + WN(t-1) + WN(t-2)
etc. so RW(t) = current shock plus all past shocks with past shocks weighted equally to the current shock so a random walk has infinite memory

44. Random Walk RW(t) -RW(t-1) = WN(t) Z0 RW(t) - Z RW(t) = WN(t)
RW(t) = 1/[1-Z] * WN(t)
RW(t) = [1 + Z + Z2 + …] WN(t)
RW(t) = WN(t) + Z WN(t) + Z2 WN(t) + ..
RW(t) = WN(t) + WN(t-1) + WN(t-2) + ...

45. Random Walk, Synthesis from White Noise
RW(t)
1/[1-Z]
WN(t)
RW(t)
1 + Z + Z
WN(t)

46. Autocovariance of RW in Theory
gRW,RW(u) = E{RW(t) - ERW(t)][RW(t-u)- ERW(t)
ERW(t) = E[WN(t) + WN(t-1) +…] = 0
gRW,RW(u=0) = E[RW(t)*RW(t)]
gRW,RW(0) = E[WN(t) + WN(t-1) + WN(t-2) + ..]* [WN(t) + WN(t-1) + WN(t-2) + ..]
gRW,RW(0) = [s2 + s2 + s2 + ….] =

47. Autocovariance of RW in Practice, Length 100
gRW,RW(u=1) = E[RW(t)*RW(t-1)]
gRW,RW(1) = E[WN(t) + WN(t-1) + WN(t-2) + ..]* [WN(t-1) + WN(t-2) + WN(t-3) + ..]
gRW,RW(1) = [s2 + s2 + s2 + ….] = 99 s2
rRW,,RW (1) = gRW,,RW(1) / gRW,,RW(0)
rRW,,RW (1) = 99/100

48. Simulated Autocorrelation, RW
Simulated Random Walk
Sample: 1 100
Included observations: 100
Autocorrelation Partial Correlation AC PAC Q-Stat Prob
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49. Part IV: Random Walk Vs. ARONE
X(t) = b*x(t-1) + wn
How close to 1 is b?

50. Exchange Rate $/Euro

51. Exchange Rate $/Euro

52. Exchange Rate, $ Per Euro
Sample: 1999: :03
Included observations: 51
Autocorrelation Partial Correlation AC PAC Q-Stat Prob
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53. Hong Kong $ Per US $

54. Hong Kong $ Per US $ Hong Kong $ Per US $ Sample: 1981:01 2003:03
Included observations: 267
Autocorrelation Partial Correlation AC PAC Q-Stat Prob
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57. Correlogram of
Ratio of inventory
To sales

60. Autocorrelation of
Residuals from an
ARONE model of
Ratio of Inventory
To Sales

61. First Difference of Ratio
Dratinvsale=ratinvsale-ratinvsale(-1)

65. Correlogram of
dratinvsale

68. Correlogram of
Residuals from
ARONE model of
First difference of
The ratio of
Inventory to sales

69. First order autoregressive
Arone(t) = b*Arone(t-1) + wn(t)
Root of the deterministic diference equation: arone(t) –b*arone(t-1) = 0
Let x1-u = arone(t-u)
x – bx0 = 0, x =b =root

70. Is b = 1? Regression: arone(t) = b* arone(t-1) + wn (t) , H0 : b=1
Equivalently, subtract arone(t-1) from both sides
(1-z)arone(t) = (b-1)*arone(t-1) + wn(t), H0 : (b-1) = 0, i.e. b=1

71. The fly in the ointment
As b approaches 1, the estimated parameter no longer has Student’s t-distribution
Dickey and Fuller use simulation to derive the appropriate distribution

72. Ratio of inventory to
Sales example of the Dickey-Fuller test