1
00:00:00,000 --> 00:00:06,074
So, Covariance stationary time series,
common structure over time, nothing

2
00:00:06,074 --> 00:00:12,145
changes.
Now, we often talk about non-stationary

3
00:00:12,145 --> 00:00:15,280
processes.
So what's the definition of a

4
00:00:15,280 --> 00:00:20,026
non-stationary process?
Well, it's something that's not stationary

5
00:00:20,026 --> 00:00:24,350
[laugh] right?
And, and you know, may, may sound, you

6
00:00:24,350 --> 00:00:30,053
know, it's, it is, it's tautological.
The reason why I say it that way is

7
00:00:30,053 --> 00:00:34,196
nonstationarity can come about in many,
many different kinds of ways.

8
00:00:34,196 --> 00:00:38,792
Remember stationarity means, you have a
common structure over time.

9
00:00:38,792 --> 00:00:43,359
If something's nonstationary, then
something is not constant over time.

10
00:00:43,359 --> 00:00:46,515
Something.
Maybe the mean's not constant over time.

11
00:00:46,515 --> 00:00:49,339
Maybe the variance is not constant over
time.

12
00:00:49,339 --> 00:00:52,670
Maybe the time dependence is not constant
over time.

13
00:00:52,670 --> 00:00:55,614
So, nonstationarity, by itself, is not
specific.

14
00:00:55,614 --> 00:00:58,493
Right?
So it's like saying you know, something is

15
00:00:58,493 --> 00:01:00,616
non-white.
Well what is it then?

16
00:01:00,616 --> 00:01:02,496
Well it's not white.
Right?

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00:01:02,496 --> 00:01:08,442
[laugh] So, in terms of nonstationery we
want anytime we say nonstationery we want

18
00:01:08,442 --> 00:01:12,968
to be specific about where the non
stationery is coming from.

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00:01:12,968 --> 00:01:19,708
Alright so here's a example.
This is what's called a determinant, de,

20
00:01:19,708 --> 00:01:24,933
deterministically trending process.
So one way that a time series maybe

21
00:01:24,933 --> 00:01:28,334
nonstationary, if the mean changes over
time.

22
00:01:28,334 --> 00:01:33,593
One way to make the mean change over time
is to explicitly make the mean a function

23
00:01:33,593 --> 00:01:37,070
of time.
So if we let Yt be beta not, plus beta one

24
00:01:37,070 --> 00:01:42,968
times time, plus epsilon t where epsilon t
is a random variable that's white noise.

25
00:01:42,968 --> 00:01:48,502
So epsilon is an uncorrelated time series
with mean zero and constant variance.

26
00:01:48,502 --> 00:01:52,467
Alright?
So what are the properties of this time

27
00:01:52,467 --> 00:01:55,978
series?
Well, if we take the expected value of Yt,

28
00:01:55,978 --> 00:02:02,477
the expected value of Yt is beta zero plus
beta one, times time, plus the expected

29
00:02:02,477 --> 00:02:05,973
value of epsilon.
But epsilon has mean zero.

30
00:02:05,973 --> 00:02:09,153
So this is beta not, plus beta one times
t.

31
00:02:09,153 --> 00:02:12,350
So, this is the mean of Yt.
Depends upon time.

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00:02:12,350 --> 00:02:14,874
So, it's not stationary.
Okay.

33
00:02:14,874 --> 00:02:22,780
Here's a plot of a deterministically
trending process, okay?

34
00:02:22,780 --> 00:02:29,577
So, it's obviously non stationary because
it's mean is an increasing function of

35
00:02:29,577 --> 00:02:32,973
time.
Any time series that has a trend, any

36
00:02:32,973 --> 00:02:37,570
discernible trend, by definition, is not
stationary, okay?

37
00:02:37,570 --> 00:02:44,282
So this is, we often will say this is
exhibiting non stationarity in the mean,

38
00:02:44,282 --> 00:02:48,550
because the mean is a function of time.
Okay?

39
00:02:48,550 --> 00:02:55,891
Now, very often when we have a
nonstationary time series, there is, a

40
00:02:55,891 --> 00:03:01,033
simple transformation of the time series
that makes it stationary.

41
00:03:01,033 --> 00:03:07,258
So, for example, if we subtract off the
deterministic trend, so we define Xt as Yt

42
00:03:07,258 --> 00:03:14,062
minus beta not, minus beta 1<i>t< /i> well
that's the light noise air term, which is</i>

43
00:03:14,062 --> 00:03:17,720
stationary.
So, de-trending a deterministically

44
00:03:17,720 --> 00:03:21,663
trending process creates a stationary time
series.

45
00:03:21,663 --> 00:03:27,874
If asset returns on average have mean
zero, then the expected gain is zero by

46
00:03:27,874 --> 00:03:34,418
investing in doing that, right?
So you know, and here the question is, is

47
00:03:34,418 --> 00:03:40,692
a mean zero a expected return a reasonable
assumption for a expected return?

48
00:03:40,692 --> 00:03:45,452
And, and here the answer depends upon the
investment Horizon.

49
00:03:45,452 --> 00:03:49,698
So, say for example your investment
horizion is one day.

50
00:03:49,698 --> 00:03:54,069
What is your expected daily return on any
investment?

51
00:03:54,069 --> 00:04:00,683
And in that case you know, essentially the
growth rate per day is typically extremely

52
00:04:00,683 --> 00:04:04,078
small and the approximation zero could
actually be very good.

53
00:04:04,078 --> 00:04:09,042
I mean you can take that even further.
Suppose you have an investment for just

54
00:04:09,042 --> 00:04:12,004
one hour.
What's the expected return on that

55
00:04:12,004 --> 00:04:14,679
investment?
You know, again, what is the expected

56
00:04:14,679 --> 00:04:18,965
price growth over any one hour period?
It's probably very, very, very small.

57
00:04:18,965 --> 00:04:22,918
On the other hand, if your investment
horizon is a year, then, you know, the

58
00:04:22,918 --> 00:04:26,600
expected return is, is most likely going
to be non zero, right?

59
00:04:26,600 --> 00:04:30,523
I mean, T bills give you at least, you
know, one percent per year and if you're

60
00:04:30,523 --> 00:04:35,016
going to invest in stocks, you would
expect to get at least better than that.

61
00:04:35,016 --> 00:04:39,092
So if your investment horizon is a longer
period of time, then you typically view

62
00:04:39,092 --> 00:04:44,061
the expected return as being positive.
Whereas if we're, the return horizon is

63
00:04:44,061 --> 00:04:48,863
quite small, then you actually might find
expected return being equal to zero to be,

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00:04:48,863 --> 00:04:52,065
be quite reasonable.
And we'll see this very clearly in the

65
00:04:52,065 --> 00:04:56,046
data as well.
Alright.

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00:04:56,046 --> 00:05:02,046
So let me continue on with a nonstationary
time series.

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00:05:02,046 --> 00:05:09,492
So another kind of non stationary time
series is, is the random walk.

68
00:05:09,492 --> 00:05:19,549
And the random walk has a, a very long
literature and holds a important, it has

69
00:05:19,549 --> 00:05:28,005
an important role in finance as well.
And so, the random walk process is a

70
00:05:28,005 --> 00:05:32,037
nonstationary process where the
nonstationarity is actually in the

71
00:05:32,037 --> 00:05:35,050
variance.
So again, something that's not stationary,

72
00:05:35,050 --> 00:05:40,032
something about the time series is
changing over time, and in the random walk

73
00:05:40,032 --> 00:05:45,015
process, we'll see what's changing over
time is the volatility of the process.

74
00:05:45,015 --> 00:05:47,009
Right?
So, what is a random walk?

75
00:05:47,009 --> 00:05:52,023
A random walk is a time series process
such that the random variable at time t is

76
00:05:52,023 --> 00:05:56,049
the random variable at time t - one plus a
white noise air term.

77
00:05:56,049 --> 00:06:00,050
So the error term has mean zero and a
constant volatility.

78
00:06:00,050 --> 00:06:02,801
Alright?
So, anybody know why a random walk is

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00:06:02,801 --> 00:06:06,729
called a random walk?
The, the term the random walk has its

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00:06:06,729 --> 00:06:10,892
origins in describing the behavior of a
drunken sailor leaving a bar.

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00:06:10,892 --> 00:06:14,561
And the idea is that the drunken sailor,
cuz he's drunk, you know?

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00:06:14,561 --> 00:06:19,472
The idea's the random walk is trying to
tell you, where's the drunken sailor going

83
00:06:19,472 --> 00:06:23,074
to be at any point in time?
And the drunken sailor, and so you think

84
00:06:23,074 --> 00:06:27,581
of epsilon as being equal you know, if
it's positive, then the, the drunken

85
00:06:27,581 --> 00:06:32,047
sailor veers off to the right.
And if epsilon is zero, then the drunken

86
00:06:32,047 --> 00:06:35,089
sailor goes forward.
And if epsilon is negative, then the

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00:06:35,089 --> 00:06:40,035
drunken sailor veers to the left.
And the idea is that, you know, these air

88
00:06:40,035 --> 00:06:45,030
terms, which represent the direction that
the sailor is walking are unpredictable.

89
00:06:45,030 --> 00:06:50,006
And so the, the drunk leaves the bar, and
can wander into the canal or, you know,

90
00:06:50,006 --> 00:06:53,046
whatever.
And so the idea's the, the evolution of

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00:06:53,046 --> 00:06:57,004
this thing is this, this random walk over
time.

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00:06:57,004 --> 00:07:01,067
And, and we think of stock prices, you
know, the idea is this would be the stock

93
00:07:01,067 --> 00:07:04,024
price today.
What, what's the stock price today?

94
00:07:04,024 --> 00:07:08,088
Well, it's what it was yesterday plus an
unpredictable component, you know, that we

95
00:07:08,088 --> 00:07:11,028
don't know.
And, and later on, we'll give an

96
00:07:11,028 --> 00:07:13,091
interpretation in the context of stock
prices.

97
00:07:13,091 --> 00:07:17,012
This could be like the random news that
hits the market.

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00:07:17,012 --> 00:07:20,078
We don't, today we don't know what the
news is going to be tomorrow.

99
00:07:20,078 --> 00:07:23,058
It could be good news, so the stock price
goes up.

100
00:07:23,058 --> 00:07:26,995
It could be neutral, so that the stock
price stays exactly where it is or it

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00:07:26,995 --> 00:07:29,770
could be bad news, the stock price could
go down.

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00:07:29,770 --> 00:07:33,556
But the idea is we can't predict what the
news is going to be.

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00:07:33,788 --> 00:07:40,489
So this is a, stochastic process, and so
let's study this a little bit and see why

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00:07:40,489 --> 00:07:46,824
it's nonstationery.
So the thing about the random walk is that

105
00:07:47,155 --> 00:07:52,018
we can do a little bit of manipulation
with it.

106
00:07:52,018 --> 00:07:58,488
So let's look at the random walk.
So Yt = Yt - one + epsilon t.

107
00:07:58,488 --> 00:08:07,207
Epsilon t is white noise.
Okay, Now we need to do recursive

108
00:08:07,207 --> 00:08:19,740
substitution.
That is, lets start, lets look at the

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00:08:19,740 --> 00:08:24,922
process at time t=1.
We know that Y1 = Y0 + E1.

110
00:08:24,922 --> 00:08:31,086
Then we're going to look at the process at
time two where Y2 = Y1 + E2.

111
00:08:31,086 --> 00:08:40,997
But notice that Y1 here is equal to Y0 +
E1.

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00:08:40,997 --> 00:08:47,784
So this process is equal to Y0 + E1 + E2.
Okay?

113
00:08:47,784 --> 00:08:56,224
And then, similarly, we go down at time T
equal, you know, capital T.

114
00:08:56,224 --> 00:09:05,335
Y at time capital T, is Y at time capital
T - one + epsilon T, and then by this

115
00:09:05,335 --> 00:09:14,800
recursive substitution, we know that this
is Y0 plus E1, plus E2, plus Et.

116
00:09:14,800 --> 00:09:18,593
So again, this is the, where the drunken
sailor is.

117
00:09:18,593 --> 00:09:21,518
This is where he is when he leaves the
bar.

118
00:09:21,518 --> 00:09:25,076
Okay?
And so where he is his first step out of

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00:09:25,076 --> 00:09:30,076
the bar is the bar door, plus the random
direction of, of where it is.

120
00:09:30,076 --> 00:09:35,487
At time two, where is the drunken sailor?
Well, it's where he was after, at time

121
00:09:35,487 --> 00:09:40,985
one, plus the random step, but where he
was at time one was leaving the bar, the

122
00:09:40,985 --> 00:09:46,228
first random step, the second random step.
And then finally, the drunken sailor at

123
00:09:46,228 --> 00:09:49,610
time capital T.
Well, it's where he was at time T - one

124
00:09:49,611 --> 00:09:53,990
plus the random step.
Well that's where he left the bar, plus

125
00:09:53,990 --> 00:09:58,337
the accumulation of all the random steps
until that point.

126
00:09:58,337 --> 00:10:04,213
So, going back to this recursive
substitution, we, in a random walk model

127
00:10:04,460 --> 00:10:09,327
we can write Y and find capital T as Y0 +
e1 + e2 up to et.

128
00:10:09,327 --> 00:10:14,636
So now we can, we can study the properties
of this process.

129
00:10:14,636 --> 00:10:25,535
So what's the expected value of Yt.
Well, it's equal to the expected value of

130
00:10:25,535 --> 00:10:32,467
y0 + e1 + e2 + et.
So that's equal to y0 plus the expected

131
00:10:32,467 --> 00:10:39,111
value of e1 plus the expected value of e2
plus the expected value of et.

132
00:10:39,111 --> 00:10:46,079
But because all the epsilons are white
noise, those are all zero, so this is

133
00:10:46,079 --> 00:10:52,517
equal to y0.
So again this is assumed to be fixed.

134
00:10:52,517 --> 00:11:00,661
So we do the random walk, we assume, we
start the random walk somewhere, right?

135
00:11:00,661 --> 00:11:06,121
This is the, the entrance to the bar,
right and so this is saying like where do

136
00:11:06,121 --> 00:11:12,076
we expect the drunken sailor to be, right
we expect the drunken sailor essentially

137
00:11:12,076 --> 00:11:17,184
because we don't, can't predict which way
he's going, we expect him to be

138
00:11:17,184 --> 00:11:23,080
essentially on a straight line outside of
the bar door right but he could, you know,

139
00:11:23,080 --> 00:11:29,000
and, and, and, so, so that's not average
you know, where the drunken sailor is.

140
00:11:32,010 --> 00:11:38,203
But you know, again, you think of, well
what does it mean to be on average where

141
00:11:38,203 --> 00:11:41,076
he is?
Think of, like putting the drunken sailor

142
00:11:41,076 --> 00:11:45,096
out of the bar, you know, 500 times, and
you let him walk, right?

143
00:11:45,096 --> 00:11:50,818
So sometimes he's going to veer off to the
left, sometimes he's going to veer off to

144
00:11:50,818 --> 00:11:54,403
the right?
What kind of on average he's going to go

145
00:11:54,403 --> 00:11:57,375
straight.
So that, that's essentially what, what

146
00:11:57,375 --> 00:12:00,544
this means.
So, lets look at the variance of this

147
00:12:00,544 --> 00:12:03,896
process.
The variance of Yy, is equal to the

148
00:12:03,896 --> 00:12:12,797
variance of, Y0 + e1 + e2 + eT.
Now, If we assume Y0 is fixed, this is

149
00:12:12,797 --> 00:12:16,773
like a constant, so this is, so this
doesn't have any variance, you know, this

150
00:12:16,773 --> 00:12:20,025
is the, this is the entrance, the entrance
to the bar.

151
00:12:20,025 --> 00:12:24,065
And then these are all the random steps.
Now, in a white noise process these are

152
00:12:24,065 --> 00:12:27,082
all uncorrelated.
Remember the variance of the sum is the

153
00:12:27,082 --> 00:12:30,094
sum of the variances plus two times all
the covariances.

154
00:12:30,094 --> 00:12:35,061
The covariances between all the air terms
are equal to zero because we have a white

155
00:12:35,061 --> 00:12:38,045
noise process.
That is the, the, the step the drunken

156
00:12:38,045 --> 00:12:42,096
sailor takes, you know, on the T-step is
independent, is uncorrelated with the next

157
00:12:42,096 --> 00:12:49,042
step and so on.
So this is equal to the variance of e1

158
00:12:49,042 --> 00:13:01,074
plus the variance of e2 plus the variance
of eT and this is because all covariances

159
00:13:01,074 --> 00:13:09,659
are equal to zero, okay?
Now, in a white noise process, each one of

160
00:13:09,659 --> 00:13:15,571
these air terms have the same variance,
sigma epsilon^2.

161
00:13:15,571 --> 00:13:22,401
So this is equal to sigma epsilon^2 +
sigma epsilon^2 + sigma epsilon^2.

162
00:13:22,401 --> 00:13:28,412
So that's T sigma epsilon^2.
So notice the, the variance.

163
00:13:28,412 --> 00:13:32,757
This is, where is the sailor after capital
T steps?

164
00:13:32,757 --> 00:13:38,376
Well, the uncertainty about where the
variance is, is equal to the number of

165
00:13:38,376 --> 00:13:41,096
steps times the, the volatility of each
step.

166
00:13:43,010 --> 00:13:51,621
So the idea here is, the more steps the
sailor takes, the less certainty we know

167
00:13:51,621 --> 00:13:58,042
where the sailor is going to be, right?
So, the volatility after one step is just

168
00:13:58,042 --> 00:14:02,073
sigma epsalon squared.
The volatility after two steps is two

169
00:14:02,073 --> 00:14:07,019
sigma epsalon squared.
The volatility after capital T steps is

170
00:14:07,019 --> 00:14:12,072
capital T times sigma epsilon squared.
So this is just saying, the further the

171
00:14:12,072 --> 00:14:17,046
sailor walks, the less certainly we know
about where the sailor is.

172
00:14:17,046 --> 00:14:23,041
So that means the volatility depends upon
time, it depends upon you know, what point

173
00:14:23,041 --> 00:14:28,046
in the sequence we have it.
So this is a process that's nonstationary

174
00:14:28,046 --> 00:14:33,799
in the variance because the variance of Yt
is t times sigma epsilon squared, it

175
00:14:33,799 --> 00:14:41,591
depends upon time.
So, and that's, that derives this

176
00:14:41,591 --> 00:14:49,041
relationship right here.
And again, maybe a picture would help.

177
00:14:50,075 --> 00:14:55,057
So if we think of Yt and t, and here is
Y0.

178
00:14:55,057 --> 00:15:04,087
And if we're doing a computer simulation.
Say, here's one process of the random

179
00:15:04,087 --> 00:15:06,001
walk.
Okay?

180
00:15:06,084 --> 00:15:12,054
And then, we simulate this thing again,
and we get other random components down

181
00:15:12,054 --> 00:15:15,805
over here.
And then you know, we do it again in, in

182
00:15:15,805 --> 00:15:21,025
as many different colors as, as, you know,
we can and so on and so forth.

183
00:15:24,005 --> 00:15:31,255
Now, the idea behind this, so if we look
at some point in time capital T.

184
00:15:31,255 --> 00:15:40,325
The volatility of the process at time t is
essentially the spread in the possible

185
00:15:40,325 --> 00:15:45,012
values of this path.
So, when t = one, you know, this process

186
00:15:45,012 --> 00:15:50,385
can't go very far, right?
But when t is a hundred then the drunken

187
00:15:50,385 --> 00:15:55,353
sailor can be, you know, way up here, or
it can be way down there.

188
00:15:55,353 --> 00:16:00,951
So the uncertainty about where he is
spreads out as time goes along.

189
00:16:00,951 --> 00:16:12,608
That's non stationary invariance.
So just like in a deterministically

190
00:16:12,608 --> 00:16:17,242
trending process, we can create a
stationary process by a simple

191
00:16:17,242 --> 00:16:20,995
transformation.
So in this random walk model, if we take

192
00:16:20,995 --> 00:16:26,389
the first difference of Yt, so if you look
at Yt, - Yt - one, by definition that's

193
00:16:26,389 --> 00:16:32,013
equal to epsilon and that's stationary.
So if we have a random walk process, you

194
00:16:32,013 --> 00:16:38,024
take the first difference of the process
and we create a stationary process and so

195
00:16:38,024 --> 00:16:42,144
the question is, you know, how important
is where you start?

196
00:16:42,356 --> 00:16:46,366
Partly it depends upon the volatility of
the steps, right?

197
00:16:46,366 --> 00:16:52,132
So if the sailor just takes tiny, tiny
little steps then you can't wander too far

198
00:16:52,132 --> 00:16:56,543
away from where he starts.
But if his volatility term is relatively

199
00:16:56,543 --> 00:17:02,079
large that the sailor can take big steps,
then eventually when you get far out in

200
00:17:02,079 --> 00:17:07,243
sequence, you're going to be pretty, you
know, mostly you're going to wander away

201
00:17:07,243 --> 00:17:12,615
from where the starting point is.
So what's true about the random walk is

202
00:17:12,615 --> 00:17:17,905
that you know, if we simulate this process
out say to infinity, it diverges to plus

203
00:17:17,905 --> 00:17:22,074
or minus infinity.
So, eventually, you know, your, your away

204
00:17:22,074 --> 00:17:25,088
from where you started with probability
one.

205
00:17:25,088 --> 00:17:29,002
And, and you've wondered off yo know, who
knows where.