Lecture 10
1
5-4-2006
I. Economic Applications of Univariate Time Series Analysis
A.. The Index of Dow Jones Industrials
There are certain puzzles for which time series analysis provides insight. For
example, recall the monthly time series for the closing value of the Dow Jones Industrials
on the last Friday of the month, DJI(t). Why doesn’t this series wander off? Why does it
not wander away from its trend line? If it is a random walk, or approximately a random
walk, we would expect it to be evolutionary and unbounded.
If one identifies this series, the value of its autocorrelation function at lag one,
ACF(1), is not significantly different from one, indicating that DJI(t) is a random walk.
The same is true of its logarithm, lnDJI(t). This conclusion is buttressed by identifying the
first difference of lnDJI(t), i.e. fractional changes in the Dow index, which is not
significantly different from white noise, WN(t), so
∆lnDJI(t) = b + WN(t),
where b is the slope of the linear trend in lnDJI(t). Recall that ∆lnDJI(t) had a positive
mean. So, from the expression above:
lnDJI(t) = lnDJI(t-1) + b + WN(t)
Recall that one of the forecasting projects was to fit a linear trend to lnDJI(t),
lnDJI(t) = a + bt + DJTRRES(t),
where DJTRRES(t) is the residual. In this model, the series lnDJI(t) which is evolutionary
can be considered as composed of two series, deterministic trend, i.e. a + bt, which is
evolutionary, and the residual series, DJTRRES(t). Thus the latter is the difference of two
evolutionary series:
Lecture 10
2
5-4-2006
lnDJI(t) - (a + bt) = DJTRRES(t).
The question is whether DJTRRES(t) is evolutionary or stationary. If it is stattionary, then
lnDJI(t) will not wander away from its trend line. If DJTRRES(t) is evolutionary then
lnDJI(t) could wander away from its trend line.
We could try to see whether DJTRRES(t) is stationary or not by identifying the
residuals, DJTRRES(t), whose ACF at lag one equals 0.904. But is this significantly
below one, indicating a stable AR(1) process.? Recall that DJTRRES(t) is from the
regression,
lnDJI(t) = a + bt + DJTRRES(t),
and we could specify the error further to be AR(1), i.e. the auxiliary equation, as
estimated, is:
DJTRRES(t) = 0.907 DJTRRES(t-1) + WN(t),
and the coefficient 0.907 with standard error 0.034. This looks promising since 0.907
appears to be significantly less than one:
H0 : b1 = 1
Ha : b1 < 1
where Student’s t-statistic under the null hypothesis is:
t = (0.907 - 1)/0.034 = -2.74.
Under the null hypothesis that b1 =1, the use of the t-statistic can be misleading and
alternative tables were developed by Fuller using simulation. The augmented DickeyFuller (ADF) test stastic indicates that b1 is significantly less than one at the 1% level. So
there is evidence for why the DOW grows at the annual rate of 11.5% and does not
wander too far from that trend.
Lecture 10
3
5-4-2006
B. Augmented Dickey-Fuller Tests
Since the standard error for estimated autocorrelations is larger than 1/√n for low
lags, such as one, relying on this standard error to test for significant differences from one
is not very satisfactory. Another specification is the regression:
∆DJTRRES(t) = d0 + d1 DJTRRES(t-1) + d2 ∆DJTRRES(t-1) + ... + U(t),
where U(t) is the residual series, presumed to be serially uncorrelated. This is the form for
the Dickey-Fuller test of whether d1 is significantly less than zero. To understand the
logic, recall the definition of the first difference:
∆DJTRRES(t) = DJTRRES(t) - DJTRRES(t-1),
and substitute for ∆DJTRRES(t) in the Dickey-Fuller regression:
DJTRRES(t) - DJTRRES(t-1) = d0 + d1 DJTRRES(t-1) + d2 ∆DJTRRES(t-1) + ... + U(t),
and adding DJTRRES(t-1) to both sides and collecting the coefficients on DJTRRES(t-1):
DJTRRES(t) = d0 + (1 + d1) DJTRRES(t-1) + d2 ∆DJTRRES(t-1) + ... + U(t),
so if d1 is significantly less than zero in the Dickey - Fuller regression, i.e.
d1 < 0,
then the coefficient on DJTRRES(t-1) will be significantly less than one,
1 + d1 < 1,
and DJTRRES(t) will be a stationary autoregressive process. The coefficient d0 can be
omitted, i.e. no constant in the Dickey-Fuller regression since DJTRRES(t) has mean
zero. The Dickey-Fuller test was conducted with no constant, no trend, and no lagged
values. Adding a lagged term did not change the conclusion. The estimate for d1 is 0.093279 with a t-statistic of -2.77, and is significantly less than zero, at the 1% level,
Lecture 10
4
5-4-2006
which has a critical value, from simulation, of -2.58. This explains why lnDJI(t) does not
wander from its trend line.
C. Unit Roots
The first order autoregressive process,
AR(t) = b1 AR(t-1) + WN(t),
or equivalently, advancing by one
AR(t + 1) = b1 AR(t) + WN(t + 1),
is a first order difference equation in its homogeneous part, i.e. setting the input WN(t) =
0:
AR(t + 1) - b1 AR(t) = 0,
and substituting xi for x(t+i), we have:
AR1 - b1 AR0 = 0 = AR - b1,
which has the root AR = b1. We have seen that an AR(1) process is stable if the absolute
value of b1 is less than one, but if b1 is one or minus one the process is evolutionary.
Hence there is interest in testing economic time series for unit roots to test for nonstationarity.
D. The Residuals, DJTRRES(t), from the Trend Model and Co-integration.
The augmented Dickey-Fuller equation for the first difference in the residuals,
DDJTRRES(t), from fitting a linear trend to DJI(t) is:
DDJTRRES(t) = d0 + d1 DJTRRES(t-1) + d2 DDJTRRES(t-1) + ... + U(t),
with an estimated value for d1 of -0.093 and a t-statistic of -2.77. This is statistically
significant and indicates the residuals are a stable AR(1) process, consistent with our
findings above from adding an AR(1) term to the regression of lnDJI(t) on trend. Thus the
Lecture 10
5
5-4-2006
difference between two evolutionary processes, lnDJI(t) and the linear trend, a + bt, is a
stable or stationary process and hence the Dow Jones Industrials will not wander away for
long from its exponential growth path.
A linear combination of two evolutionary processes, where the combination is
stationary, is said to be co-integrated. In this case the linear combination weights, i.e.
integrating factors, are (1, -1). The two evolutionary processes, lnDJI(t) and trend are said
to be integrated of order one, while their difference, lnDJI(t) - trend, is integrated of order
zero.
E. Error Correction Models
The Dickey-Fuller equation for the residuals, DJTRRES(t), can be interpreted in
terms of error correction. For example, substituting lnDJI(t-1) - trend(t-1), for
DJTRRES(t-1) in the equation above:
DDJTRRES(t) = d0 + d1 [DJI(t-1) - trend(t-1)]+ d2 DDJTRRES(t) + ... + U(t).
If the lnDJI(t) wandered too far above the trend line last period, i.e. the residual, ,
DJTRRES(t-1), was large last period, then the effect of the negative coefficient d1 will be
to decrease the change in residuals, DDJTRRES(t), this period and bring the DJI(t) back
closer to the trend line.
F. Puchasing Power Parity
The real exchange rate between two currencies is the ratio of the nominal
exchange rate, for example the $ price of the Yen, divided by an index of the relative
price of a bundle of commodities in the two countries. Taking natural logarithms, the
natural logarithm of the real exchange rate, R(t), will equal the difference between the
Lecture 10
6
5-4-2006
natural logarithm of the nominal exchange rate, S(t), and the natural logarithm of the
relative price index, P(t);
R(t) = S(t) - P(t).
Economists have found unit roots in the time series for nominal exchange rates
and relative price indices, indicating that these series are evolutionary. Does this imply
that the real exchange rate is also evolutionary. The concept of puchasing power
parity(PPP) is that the real exchange rate should be stationary and not go wandering off.
Also, under purchasing power parity, the nominal exchange rate and the relative price
index should move together, i.e.:
S(t) = a + b P(t) + U(t),
where U(t) is a residual process and b=1 if S(t) and P(t) move together, in which case,
S(t) - P(t) = a + U(t).
This form was used in earlier studies to test for PPP.
The augmented Dickey-Fuller regression for unit root tests in the real exchange
rate is:
∆R(t) = d0 + d1 R(t-1) + d2 ∆R(t-1) + ... + e(t),
where the residual process is presumed to be serially uncorrelated.
The relationship between R(t) and S(t) and P(t) can be used to substitute for R(t)
in this augmented Dickey-Fuller equation:
∆{S(t) - P(t)} = d0 + d1 {S(t-1) - P(t-1)} + d2 ∆{S(t-1) - P(t-1)} + ... + e(t),
or, rearranging:
∆S(t) = d0 + ∆ P(t) + d1 {S(t-1) - P(t-1)} + d2 ∆{S(t-1) - P(t-1)} + ... + e(t).
Lecture 10
7
5-4-2006
Note that this places restrictions on the parameters: the coefficient on ∆ P(t) is one, and
the coefficient, d1 , on S(t-1) equals the coefficient on P(t-1). Hence Dickey-Fuller tests
on the real exchange rate may be testing, in fact, these parameter restrictions, and not just
whether the real exchange rate has a unit root.
Lastly, note that if S(t) and P(t) are co-integrated with cointegrating factors (1, -1),
then the real exhange rate could be stationary and PPP may be true.