2. Time Series Data
• One of the important and frequent types of data used in
empirical analysis.
• But it poses several challenges to
econometricians/practitioners. E.g.
1. Empirical work based on time series data assumes that the
underlying time series is stationary.
2. Autocorrelation: because the underlying time series data is
non-stationary.
3. Spurious/nonsense regression: a very high R2 and significant
regression coefficients (though there is no meaningful
relationship between the two variables)
3. Key Concepts
1. Stochastic Processes
i. Stationarity Processes
ii. Purely Random Processes
iii. Non-stationary Processes
2. Random Walk Models
i. Random Walk with Drift
ii. Random Walk without Drift
3. Unit Root Stochostic Processes
4. Deterministic and Stochastic Trends
5. The Phenomenon of Spurious Regression
6. Tests of Stationarity/non-stationarity
i. Graphical Method
ii. Unit Root Tests
4. 1) Stochastic Processes
• Stochastic (Random) Process: collection of
random variables ordered in time.
o NOTATIONS: Let Y a random variable, Y(t) if continuous
(e.g. electrocardiogram), and Yt if discrete (e.g. GDP, PDI,
etc.).
o Now, If we let Y represent GDP, then we can have Y1, Y2, Y3,
... , Y20 where the subscript 1 denotes the 1st observation
(i.e. GDP for the 1st quarter of 1st year) and the subscript 20
denotes the last observation (i.e. GDP for the 4th quarter of
5th year).
5. (1) Stochastic Processes
• Stationary Stochastic Processes: A stochastic process is
said to be stationary/ weakly /covariance/2nd-order
stationary if:
o Its mean and variance are constant over time, and
o The value of the covariance between the two time periods depends only
on the distance/lag between the two time periods and not the actual time
at which the covariance is computed.
o E.g. let’s Yt be a stochastic process, then;
– Mean: E(Yt ) = µ …………………………………………..
(1)
– Variance: var (Yt ) = E(Yt − µ)2 = σ2 ………………………………..
(2)
– Covariance: γk = E[(Yt − µ)(Yt +k − µ)] ……………..…………(3)
• Where γk, the covariance (or auto-covariance) at lag k,
• If k = 0, we obtain γ0, which is simply the variance of Y (= σ2); if k =
1, γ1 is the covariance between two adjacent values of Y
6. Why are Stationary Time Series so
Important?
• Because if a time series is non-stationary, we can study its
behavior only for the time period under consideration, and as a
consequence, it is not possible to generalize it to other time
periods.
• Therefore, for the purpose of forecasting, such (non-stationary)
time series may be of little practical value.
• Non-stationary Stochastic Processes: Although our
interest is in stationary time series, one often encounters non-
stationary time series
• A non-stationary time series will have a time-varying mean
or a time-varying variance or both.
7. • We call a stochastic process(time series) purely random/white noise
process if it has zero mean, constant variance σ2, and is serially
uncorrelated i.e. [ut ∼ IIDN(0, σ2)].
• Note: Here onward, in all equations the assumption of “white noise” will be
applicable on ut .
• Yt= a +b1xt+b2xt+et
White Noise Processes
8. 2) Random Walk Model (RWM)
• The classic example of non-stationary time series is the
Random Walk Model (RWM).
• It is often said that asset prices, such as stock prices or
exchange rates, follow a random walk (i.e. non-
stationary).
• Types of Random Walks:
a) Random Walk Without Drift:
GDP=bGDPt-1+ut
i.e. no constant/intercept term and
b) Random Walk With Drift
i.e. a constant term is present
GDP= a+bGDPt-1+ut
9. a) Random Walk without Drift
• The time series Yt is said to be a random walk without drift, if
Yt = Yt −1 + ut ……. (4)
• Here, the value of Y at time (t) is equal to its value at time (t − 1)
plus a random shock; thus it is an AR(1) model.
• Believers in the Efficient Capital Market Hypothesis
argue that stock prices are essentially random and therefore
there is no scope for profitable speculation in the stock market:
• If one could predict tomorrow’s price on the basis of today’s
price, we would all be millionaires.
10. Further Explanation
• Now from Yt = Yt −1 + ut……. (4) we can write:
Y1 = Y0 + u1
=> Y2 = Y1 + u2 = Y0 + u1 + u2 (u1+u2=vt)
=> Y3 = Y2 + u3 = Y0 + u1 + u2 + u3 and so on…
• In general, if the process started at some time 0 with a value of Y0, we
have: Yt = Y0 +∑ut ……..
……………………….(5)
• Therefore,
E(Yt ) = E(Y0 + ∑ut) = Y0 (why?) ………… (6)
Because, ut is “white noise”
• In like fashion, it can be shown that: var (Yt ) = tσ2 …… (7)
• Thus, the mean of Y is equal to its starting value, which is constant, but
as t increases, its variance increases indefinitely (thus violating the
condition of stationarity).
• In short, the RWM without drift is a non-stationary stochastic process.
11. Further Explanation (cont.)
• Now, if you write Yt = Yt −1 + ut….. (4) as
(Yt − Yt−1) = ∆Yt = ut ……. (8)
• It shows that, while Yt is non-stationary, its 1st
difference is stationary.
• In other words, the 1st differences of a random walk
time series are stationary.
12. b) Random Walk with Drift
• Let’s modify, Yt = Yt −1 + ut…………….(4) as
follows:
Yt = δ + Yt−1 + ut….....………….(9)
where δ is the drift parameter.
• The name drift comes from the fact that if we write
the preceding equation as:
Yt − Yt−1 = ∆Yt = δ + ut …………. (10)
• It shows that Yt drifts upward/downward, depending
on δ being positive/negative.
13. Further Explanation
• Note that model Yt = δ + Yt−1 + ut ………….. (9) is also an AR(1)
model.
• Following the procedure discussed for Random Walk Without Drift,
it can be shown that for the random walk with drift model (9),
E(Yt ) = Y0 + t · δ ……….. (11)
var (Yt ) = tσ2 …………….... (12)
• Here, again for RWM with drift the mean as well as the variance
increases over time, again violating the conditions of stationarity.
• In short, RWM, with or without drift, is a non-stationary stochastic
process.
• The random walk model is an example of what is known in the
literature as a Unit Root Process.
14. 3) Unit Root Stochastic Process
• Let’s write the RWM Yt = Yt −1 + ut….. (4) as:
Yt = ρYt−1 + ut − 1 -1 ≤ ρ ≤ 1 …….(13)
• If ρ = 1, (13) becomes a RWM (without drift).
• If ρ is in fact 1, we face what is known as the unit root problem
(non-stationarity); as the variance of Yt is not stationary.
• The name unit root is due to the fact that ρ = 1.
• Thus the terms non-stationarity, random walk, and unit root can
be treated as synonymous.
• If, however, |ρ| ≤ 1, then the time series Yt is stationary in the
sense we have defined it.
• Note: Unit Root Stochastic Process will be further explained in
Unit Root Test of Stationarity.
15. Trend Stationary (TS) and Difference Stationary
(DS)
Stochastic Processes
• Deterministic Trend: if the trend in a time series is completely
predictable
• Stochastic Trend: if it is not predictable
• E.g. consider the following model of the time series Yt .
Yt = β1 + β2t + β3Yt−1 + ut ………………….(14) (TSP)
• Now we have the following possibilities:
1. Pure Random Walk: If in (14) β1 = 0, β2 = 0, β3 = 1, we get:
Yt = Yt−1 + ut …………(15)
• Which is nothing but a RWM without drift and is therefore non-stationary.
• But note that, if we write (2.2) as
∆Yt = (Yt − Yt−1) = ut ………… (8)
• It becomes stationary, as noted before.
• Hence, a RWM without drift is a Difference Stationary Process (DSP).
16. Trend Stationary (TS) and Difference Stationary
(DS)
Stochastic Processes (cont.)
2. Random Walk With Drift: If in Yt = β1 + β2t + β3Yt−1 + ut
……(14) β1 ≠ 0, β2 = 0, β3 = 1, we get:
Yt = β1 + Yt−1 + ut ……..(16)
• which is a random walk with drift and is therefore, non-
stationary.
• If we write it as: (Yt − Yt−1) = ∆Yt = β1 + ut ……..(16a)
• this means Yt will exhibit a positive (β1 > 0) or negative (β1 < 0)
trend.
• Such a trend is called a Stochastic Trend.
• Equation (16a) is a DSP process because the non-stationarity in
Yt can be eliminated by taking first differences of the time series.
17. Trend Stationary (TS) and Difference Stationary
(DS)
Stochastic Processes (cont.)
3. Deterministic Trend: If in Yt = β1 + β2t + β3Yt−1 + ut ……(14)
β1 ≠ 0, β2 ≠ 0, β3 = 0, we obtain:
Yt = β1 + β2t + ut ………..(17)
• Which is called a trend stationary process (TSP).
• Although the mean of Yt is β1 + β2t, which is not constant, its
variance ( = σ2) is.
• Once the values of β1 and β2 are known, the mean can be
forecast perfectly.
• Therefore, if we subtract the mean of Yt from Yt, the resulting
series will be stationary, hence the name trend stationary.
18. Trend Stationary (TS) and Difference Stationary
(DS)
Stochastic Processes (cont.)
4. Random Walk With Drift and Deterministic Trend: If in Yt =
β1 + β2t + β3Yt−1 + ut ……(14) β1 ≠ 0, β2 ≠ 0, β3 = 1, we obtain:
Yt = β1 + β2t + Yt−1 + ut ……….…..(18)
• We have a random walk with drift and a deterministic trend,
which can be seen if we write this equation as:
∆Yt = β1 + β2t + ut ……….……….(18a)
• Which means that Yt is non-stationary.
5. Deterministic Trend With Stationary AR(1) Component: If
in Yt = β1 + β2t + β3Yt−1 + ut ……(14) β1 ≠ 0, β2 ≠ 0, β3 < 1, then
we get:
Yt = β1 + β2t + β3Yt−1 + ut ..……….(19)
• Which is stationary around the deterministic trend.
19. The Phenomenon of Spurious Regression
• Stationary Time Series are important, consider the
following two random walk models:
Yt = Yt−1 + ut ………….. (20)
Xt = Xt−1 + vt ………….. (21)
• Where we generated 500 observations of ut from ut ∼ N(0, 1)
and 500 observations of vt from vt ∼ N(0, 1) and assumed that
the initial values of both Y and X were zero.
• We also assumed that ut and vt are serially uncorrelated as well
as mutually uncorrelated.
• Both these time series are non-stationary; i.e. they are I(1) or
exhibit stochastic trends.
20. The Phenomenon of Spurious
Regression (cont.)
• Suppose we regress Yt on Xt.
• Since Yt and Xt are uncorrelated I(1) processes, the
R2 from the regression of Y on X should tend to zero;
that is, there should not be any relationship between
the two variables.
• But wait till you see the regression results:
21. The Phenomenon of Spurious
Regression (cont.)
• As you can see, the coefficient of X is highly
statistically significant, and, although the R2 value is
low, it is statistically significantly different from zero.
• From these results, you may be tempted to conclude
that there is a significant statistical relationship
between Y and X, whereas a priori there should be
none.
• This is the phenomenon of spurious/non-sense
regression, first discovered by Yule.
22. The Phenomenon of Spurious
Regression (cont.)
• Yule showed that (spurious) correlation could
persist in non-stationary time series even if the
sample is very large.
• That there is something wrong in the preceding
regression is suggested by the extremely low
Durbin–Watson d value, which suggests very
strong first-order autocorrelation.
23. The Phenomenon of Spurious Regression
(cont.)
• R2 > d is a good rule of thumb to suspect that the
estimated regression is spurious, as in the example
above.
• This example is a strong reminder that one should be
extremely cautious of conducting regression analysis
based on time series that exhibit stochastic trends.
24. Tests of Stationarity
• In practice we face two important questions:
– How do we find out if a given time series is
stationary or not?
– Is there a way that it can be made stationary?
• Prominently discussed tests in the literature are:
– Graphical Analysis
– The Unit Root Test
25. 1. Graphical Analysis
• Before pursuing a formal test, it is always
advisable to plot the time series under study
• E.g. take the GDP time series.
• You will see that over the period of study GDP
has been increasing (i.e. showing an upward
trend)
• This perhaps suggests that the GDP series is
not stationary (also more or less true of the
other economic time series).
26. 2- The Unit Root Test
• The widely popular test of stationarity is the unit root
test.
• We start with: Yt = ρYt−1 + ut − 1 ≤ ρ ≤ 1 (1.1)
Where ut is a white noise error term.
• We know that if ρ = 1 (i.e. in the case of the unit root)
(1.1) becomes a Random Walk Model Without Drift,
(a non-stationary stochastic process).
• Therefore, why not simply regress Yt on its lagged
value Yt−1 and find out if the estimated ρ is statistically
equal to 1? If it is, then Yt is non-stationary.
27. 2- The Unit Root Test (cont.)
• Procedure:
Yt = ρYt−1 + ut (1.1)
=> Yt − Yt−1 = ρYt−1 − Yt−1 + ut (4.1)
=> Yt − Yt−1 = Yt−1 (ρ − 1)+ ut or
=> Yt − Yt−1 = (ρ − 1)Yt−1 + ut
which can be alternatively written as:
=> ∆Yt = δYt−1 + ut (4.2)
where δ = (ρ − 1) and ∆= first-difference (Yt − Yt−1 )
28. 2- The Unit Root Test (cont.)
• Hence, to estimate (4.2) and test the H0: δ = 0.
• If δ = 0, then ρ = 1 (i.e. unit root/time series under
consideration is non-stationary) and (21.9.2) will
become ∆Yt = (Yt − Yt−1) = ut (4.3)
• Since ut is a white noise error term, it is stationary,
which means that the first differences of a random
walk time series are stationary.
• Now let’s turn to the estimation of (4.2).
29. 2- The Unit Root Test (cont.)
• We have to take the first differences of Yt and regress
them on Yt−1 and see if the estimated slope co-efficient in
this regression (δ) is zero or not.
• If it is zero, we conclude that Yt is non-stationary.
• But if it is negative, we conclude that Yt is stationary.
• The only question is which test we use to find out if the
estimated co-efficient of Yt−1 in (4.2) is zero or not?
• Unfortunately, under the null hypothesis that δ = 0 (i.e., ρ
= 1), the t value of the estimated coefficient of Yt−1 does
not follow the t distribution even in large samples; i.e.
it does not have an asymptotic normal distribution.
30. What is the alternative?
• Dickey and Fuller have shown that under the null
hypothesis that δ = 0, the estimated t-value of
the coefficient of Yt−1 in (4.2) follows the τ (tau)
statistic.
• These authors have computed the critical values
of the tau statistic on the basis of Monte Carlo
simulations.
• Interestingly, if the hypothesis that δ = 0 is
rejected (i.e. the time series is stationary), we can
use the usual t test.
31. DF Test
• The actual procedure of implementing the DF test
involves several decisions.
• In discussing the nature of the unit root process in
Sections 21.4 and 21.5, we noted that a random
walk process may have no drift, or it may have
drift or it may have both deterministic and
stochastic trends.
• To allow for the various possibilities, the DF test
is estimated in three different forms, i.e. under
three different null hypotheses.
32. • Yt is a random walk: Yt = δYt−1 + ut (4.2)
• Yt is a random walk with drift: Yt = β1 + δYt−1 + ut
(4.4)
• Yt is a random walk with drift around a stochastic
trend: Yt = β1 + β2t + δYt−1 + ut (4.5)
Where t is the time or trend variable.
33. • In each case, the null hypothesis is that δ = 0;
i.e. there is a unit root—the time series is non-
stationary.
• The alternative hypothesis is that δ is less than
zero; i.e. the time series is stationary.
• If the null hypothesis is rejected, it means that Yt
is a stationary time series with zero mean in the
case of (4.2), that Yt is stationary with a nonzero
mean [= β1/(1 − ρ)] in the case of (4.4), and that
Yt is stationary around a deterministic trend in
(4.5).
34. • It is extremely important to note that the critical values of
the tau test to test the hypothesis that δ = 0, are different for
each of the preceding three specifications of the DF test,
which can be seen clearly from Appendix D, Table D.7.
• Moreover, if, say, specification (4.4) is correct, but we
estimate (4.2), we will be committing a specification error,
whose consequences we already know from Chapter 13.
• The same is true if we estimate (4.4) rather than the true
(4.5).
• Of course, there is no way of knowing which specification
is correct to begin with.
• Some trial and error is inevitable, data mining
notwithstanding.
35. • The actual estimation procedure is as follows:
• Estimate (4.2), or (4.3), or (4.4) by OLS; divide the
estimated coefficient of Yt−1 in each case by its standard
error to compute the (τ) tau statistic; and refer to the DF
tables (or any statistical package).
• If the computed absolute value of the tau statistic (|τ |)
exceeds the DF or MacKinnon critical tau values, we reject
the hypothesis that δ = 0, in which case the time series is
stationary.
• On the other hand, if the computed |τ | does not exceed the
critical tau value, we do not reject the null hypothesis, in
which case the time series is non-stationary.
• Make sure that you use the appropriate critical τ values.
36. • Let us return to the U.S. GDP time series. For
this series, the results of the three regressions
(4.2), (4.4), and (4.5) are as follows:
• The dependent variable in each case is Yt =
GDPt
37. • GDPt = 0.00576GDPt−1…… (4.6)
• t = (5.7980) R2 =−0.0152 d = 1.34
• GDPt = 28.2054 − 0.00136GDPt−1…… (4.7)
• t = (1.1576) (−0.2191) R2 = 0.00056 d = 1.35
• GDPt = 190.3857 + 1.4776t − 0.0603GDPt−1
• t = (1.8389) (1.6109) (−1.6252)…….. (4.8)
• R2 = 0.0305 d = 1.31
38. • Our primary interest here is in the t ( = τ) value of
the GDPt−1 coefficient.
• The critical 1, 5, and 10 percent τ values for
model (4.6) are −2.5897, −1.9439, and −1.6177,
respectively, and are −3.5064, −2.8947, and
−2.5842 for model (4.7) and −4.0661, −3.4614,
and −3.1567 for model (21.3.8).
• As noted before, these critical values are different
for the three models.
39. • Before we examine the results, we have to decide
which of the three models may be appropriate.
• We should rule out model (4.6) because the
coefficient of GDPt−1, which is equal to δ is positive.
• But since δ = (ρ − 1), a positive δ would imply that
ρ> 1.
• Although a theoretical possibility, we rule this case
out because in this case the GDP time series would be
explosive.
40. • That leaves us with models (4.7) and (4.8). In both
cases the estimated δ coefficient is negative, implying
that the estimated ρ is less than 1.
• For these two models, the estimated ρ values are
0.9986 and 0.9397, respectively.
• The only question now is if these values are
statistically significantly below 1 for us to declare
that the GDP time series is stationary.
41. • For model (4.7) the estimated τ value is −0.2191, which in
absolute value is below even the 10 percent critical value of
−2.5842.
• Since, in absolute terms, the former is smaller than the
latter, our conclusion is that the GDP time series is not
stationary.
• The story is the same for model (4.8).
• The computed τ value of −1.6252 is less than even the 10
percent critical τ value of −3.1567 in absolute terms.
• Therefore, on the basis of graphical analysis, the
correlogram, and the Dickey–Fuller test, the conclusion is
that for the quarterly periods of 1970 to 1991, the U.S. GDP
time series was non-stationary; i.e., it contained a unit root.
42. • Ho: Data follow non-stationariety or follow
Random Walk or Unite root
• H1: Data is stationariety.
• Tau cal> tautab= We reject Ho and accept H1.
i.e data follow stationary pattern
• Taucal<tautab= we failed to reject Ho i.e we
accept Ho and reject H1.
