ECON3350/7350 Deterministic and Stochastic Trends
Eric Eisenstat
The University of Queensland
Lecture 5
Eric Eisenstat
(School of Economics) ECON3350/7350 Week 5 1 / 23
Multiple Time Series Models
Recommended readings
Author
Title
Chapter
Call No
Enders Verbeek
Applied Econometric Time Series, 4e
A Guide to Modern Econometrics
4 8.2-8.5
HB139 .E55 2015 HB139 .V465 2012
Eric Eisenstat (School of Economics) ECON3350/7350 Week 5 2 / 23
Trends
Many (not all) economic time series appear to have a trend in the mean and/or variance.
9.6 9.5 9.4 9.3 9.2 9.1 9.0 8.9 8.8
1975 1980
1985 1990
1995 2000
2005 2010
Figure: Australia log real GDP per capita.
What type of trend is present in this data, and how can we transform it to
stationary data?
Eric Eisenstat (School of Economics) ECON3350/7350 Week 5 3 / 23
Trend Stationary
A trend can be modeled as deterministic, that is a known function (up to estimated parameters) of t. This is consistent with exogenous growth theories.
For example, a linear trend is specified by
yt =a0 +δt+a1yt−1 +εt, |a1|<1.
The trend can be “taken out” by setting zt = yt − δ/(1 − a1)t, which implies
zt =a0 +a1zt−1 +εt, |a1|<1.
zt is a stationary AR(1) process; δ/(1 − a1) is the long-run growth rate.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 5
4 / 23
De-trended Data
In practice, remove a linear trend by regressing yt on a constant and t, then save the residuals (this will also de-mean the data).
.08 .06 .04 .02 .00 -.02 -.04 -.06 -.08
Autocorrelation
Autocorrelation
1975 1980
1985 1990 1995
2000 2005 2010
Eric Eisenstat
(School of Economics)
ECON3350/7350
Week 5
5 / 23
Levels
Detrended
Difference Stationary
Real Business Cycle theories suggestion innovations (i.e. productivity shocks) have a permanent effect on productivity.
Endogenous growth theories imply an alternative form of trend, e.g. arisingfromδ=0anda1 =1:
yt = a0 + yt−1 + εt.
In this setting, the data contains a stochastic trend, where the expected
change is E(∆yt) = a0; hence, a0 is the long-run growth rate. A stochastic trend is removed simply by differencing the data.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 5 6 / 23
Difference Operator
The difference operator is defined in terms of the lag operator as ∆ = 1−L. Hence,
∆yt =(1−L)yt =yt −yt−1.
As with lag operators in general, we can combine two difference operator
to construct a difference-in-difference operator, i.e. ∆2yt = ∆(∆yt) = (1 − L)(yt − yt−1)
= (yt − yt−1) − (yt−1 − yt−2) = ∆yt − ∆yt−1.
In general,
∆d = (1 − L)dyt = (1 − L)((1 − L)d−1yt) = ∆d−1yt − ∆d−1yt−1.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 5
7 / 23
Difference Operator
Other notation you might come across:
∆1yt =(1−L)yt =yt −yt−1 =∆yt, ∆2yt =(1−L2)yt =yt −yt−2 ̸=∆2yt,
.
∆syt =(1−Ls)yt =yt −yt−s ̸=∆syt. Finally, we can mix these to obtain:
∆ds =(1−Ls)dyt =∆s···∆syt i.e.,applieddtimes.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 5 8 / 23
Stochastic Trends
Recall that in the stationary AR(1) with yt =a0 +a1yt−1 +εt, |a1|<1,
=a0(1+a1 +···+at−1)+at1y0 +εt +a1εt−1 +···+at−1ε1, 11
the effect of a shock dies out as t increases.
In the presence of a stochastic trend characterized by a1 = 1,
yt = a0 + yt−1 + εt,
=a0t+y0 +εt +εt−1 +···+ε1,
all past shocks have permanent effects.
The AR(1) with a0 = 0 and a1 = 1 is called a random walk (RW).
Intuitively, starting from y0 and for large enough t, the random
variable yt can attain any value with equal probability.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 5 9 / 23
Properties of the Random Walk
The RW does not have a finite unconditional expectation, but it does have the following conditional expectations:
E(yt | yt−1) = yt−1, E(yt | y0) = y0.
Therefore, the mean is constant conditional on a particular initial value y0. Conditional on an initial value y0, the variance for a RW is:
E((yt −y0)2|y0)=Var(εt +···+ε1|y0)=tσε2, and the covariance for observations s periods apart is:
E((yt − y0)(yt−s − y0) | y0) = (t − s)σε2.
The variance and covariances are not constant ⇒ the RW is not stationary.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 5 10 / 23
Random Walk with Drift
The AR(1) with a0 ̸= 0 and a1 = 1 is called a random walk with a drift (RW).
The sign of a0 indicates the direction that yt will drift over time; however, the actual trend is influenced by the shocks in each period.
The moments conditional on the initial value y0 are:
E(yt | y0) = a0t + y0,
Var(yt|y0)=Var(a0t+y0 +εt +···+ε1|y0)=tσε2,
Cov((yt, yt−s | y0) = (t − s)σε2. Therefore, all moments depend on t.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 5
11 / 23
Characteristic Roots
For more general processes with p > 1, stationarity is captured by the characteristic equation:
a(z)=1−a1z−···−apzp =0. (1) Here, z is a complex number, i.e. z ∈ C. The roots of a(z)—that is, all
values of z that solve (1)—are called the characteristic roots of the system.
The characteristic roots control the dynamic responses of yt to any shock. If all |z| > 1, then the effect of a shock vanishes over time.
If there exists at least one |z| = 1, then the effect of a shock persists over time.
If there exists at least one |z| < 1, then the effect of a shock increases over time.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 5 12 / 23
Unit Root Processes
For the AR(1), the characteristic equation is a(z) = 1 − a1z = 0, and the only root is z = 1/a1.
a1 = 1 corresponds to z = 1, which we call the unit root. FortheAR(p),thelagpolynomiala(L)=1−a1L−···−apLp canbe
regarded as a polynomial function a(z), z ∈ C, and factored as a(z) = (1 − α1z)(1 − α2z) · · · (1 − αpz),
where α1,...,αp are functions of a1,...,ap.
If any αj = 1 for j = 1,...,p, then the process will have a unit root. If
there are d such αj = 1, then the process will have d unit roots.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 5 13 / 23
AR(2) Example
For the model
yt = a1yt−1 + a2yt−2 + εt,
the characteristic polynomial is a(z) = 1 − a1z − a2z2 = 0.
Completing the square (or using the quadratic formula), we can factor a(z) as
2a 2a 11
a(z)= 1+a +a2+4a z 1+a −a2+4a z . 221 221
The necessary and sufficient conditions for stationarity are:
a1 +a2 <1, −a1 +a2 <1, −a2 <1.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 5 14 / 23
Difference Stationary or Integrated Processes
If in an AR(p) there exists one unit root (e.g. αj = 1), then the first difference is an AR(p − 1). Factoring a(z) and substituting L for z we get:
a(L) = (1 − α1L) · · · (1 − αj−1L)(1 − αj+1L) · · · (1 − αpL)(1 − L) = (1 − b1L − · · · − bp−1Lp−1)(1 − L) = b(L)(1 − L).
If the remaining p − 1 roots are greater than one in absolute value, then the process is difference stationary or integrated of order one, denoted I(1).
More generally, if an AR(p) in yt contains d unit roots, then ∆dyt is an AR(p − d), and if all remaining roots are greater than one in absolute value, then the process is I(d).
Eric Eisenstat (School of Economics) ECON3350/7350 Week 5 15 / 23
Dickey-Fuller Test
Dickey Fuller tests for unit roots in an AR(1), i.e. (a1 = 1). To implement the test, re-arrange the model as
yt = a1yt−1 + εt, yt−yt−1 = a1yt−1−yt−1 + εt,
∆yt = γyt−1 + εt,
where γ = a1 − 1.
Test H0 : γ = 0 against H1 : γ1 < 0, with critical values tabulated under
the Dickey-Fuller distribution.
What about testing for unit roots in an AR(p)?
Eric Eisenstat (School of Economics) ECON3350/7350 Week 5 16 / 23
Augmented Dickey-Fuller Test
For the AR(2), follow a similar strategy to obtain
yt = a1yt−1 + a2yt−2 + εt,
yt−yt−1 = a1yt−1−yt−1 + a2yt−2+a2yt−1 − a2yt−1 + εt,
∆yt = γyt−1 + b1∆yt−1 + εt, where γ = a1 + a2 − 1 and b1 = −a2.
In the AR(2), a unit root exists iff a1 + a2 = 1 or equivalently γ = 0. For the general AR(p) case, estimate
∆yt = γyt−1 + b1∆yt−1 + · · · + bp−1∆yt−p+1 + εt, and test γ = 0.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 5 17 / 23
Including Deterministic Regressors and Trends
yt =a1yt−1 +···+apyt−p +εt,
⇔∆yt =γyt−1 +b1∆yt−1 +···+bp−1∆yt−p+1 +εt,
yt =a0 +a1yt−1 +···+apyt−p +εt,
⇔∆yt =a0 +γyt−1 +b1∆yt−1 +···+bp−1∆yt−p+1 +εt,
yt =a0 +δt+a1yt−1 +···+apyt−p +εt,
⇔∆yt =a0 +δt+γyt−1 +b1∆yt−1 +···+bp−1∆yt−p+1 +εt.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 5 18 / 23
Augmented Dickey-Fuller Test
To implement the Augmented Dickey-Fuller (ADF), estimate the most general specification:
∆yt =a0 +δt+γyt−1 +b1∆yt−1 +···+bp−1∆yt−p+1 +εt.
Totestforaunitroot,useH0 :γ=0againstH1 :γ<0. Thisistheττ test, with critical values tabulated in Enders (Table A); software provides p-values.
interpretation: under H0, the process is I(1) with a drift and deterministic trend (i.e., exhibits linear growth); under H1, it is trend stationary.
To test for both a unit root and deterministic trend, use H0 : δ = 0, γ = 0 against H1 : δ ̸= 0, γ < 0. This is the φ3 test, with critical values tabulated in Enders (Table B).
interpretation: under H0, the process is I(1) with a drift (i.e., exhibits
constant growth); under H1, it is trend stationary.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 5 19 / 23
Other Variants of the ADF
OmittingδtandtestingH0 :γ=0againstH1 :γ<0leadstotheτμ test (critical values in Enders, Table A).
OmittingδtandtestingH0 :a0 =0,γ=0againstH1 :a0 ̸=0,γ<0 leads to the φ1 test (critical values in Enders, Table B).
Omittinga0+δtandtestingH0 :γ=0againstH1 :γ<0leadstotheτ test (critical values in Enders, Table A).
All ADF test statistics have non-standard distributions.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 5 20 / 23
ADF Test: Example 1
Null Hypothesis: LGDPPC has a unit root
Exogenous: Constant, Linear Trend
Lag Length: 1 (Automatic - based on SIC, maxlag=13)
Augmented Dickey-Fuller test statistic Test critical values: 1% level 5% level
10% level
*MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation Included observations: 145 after adjustments
t-Statistic
−2.856187 −4.022586 −3.441111 −3.145082
Prob.*
0.1801
Variable Coefficient LGDPPC(-1) −0.084277
Std. Error
0.029507 0.081826 0.261091 0.000143
t-Statistic
−2.856187 0.747723 2.868583 2.910159
Prob.
0.0049 0.4559 0.0048 0.0042
0.004185
0.009582 −6.468984 −6.386867 −6.435617
1.987599
D(LGDPPC(-1))
C @TREND(”1973Q3”)
R-squared
Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)
0.061183 0.748960 0.000416
0.057617 0.037566 0.009400 0.012459 473.0013 2.873541 0.038485
Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat
Eric Eisenstat
(School of Economics)
ECON3350/7350
Week 5
21 / 23
ADF Test: Example 2
Null Hypothesis: LGDPPC has a unit root
Exogenous: Constant
Lag Length: 1 (Automatic - based on SIC, maxlag=13)
Augmented Dickey-Fuller test statistic Test critical values: 1% level 5% level
10% level
*MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation Included observations: 145 after adjustments
t-Statistic
0.217256 −3.475819 −2.881400 −2.577439
Prob.*
0.9729
Variable Coefficient LGDPPC(-1) 0.000858
D(LGDPPC(-1)) 0.024860 C −0.003823
R-squared 0.001013
Std. Error
t-Statistic
Prob.
0.8283 0.7649 0.9164
0.004185
0.009582 −6.424448 −6.362860 −6.399423
1.970315
Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)
−0.013057 0.009644 0.013208 468.7725 0.072011 0.930555
0.003950 0.217256 0.082968 0.299638 0.036359 −0.105155
Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat
Eric Eisenstat
(School of Economics)
ECON3350/7350
Week 5
22 / 23
Summary
Many economic variables appear to be nonstationary. e.g., trending behaviour in real per capita GNP wandering behaviour in interest rates
The type of nonstationary process has important economic implications.
deterministic trends: predictable behaviour with no long-run memory (shocks “die out”)
stochastic trends (& unit roots): trend in variance implying growing unpredictability in the future, permanent effect of shocks, complete long-run memory
Trend stationary vs. difference stationary processes.
It is important for economic reasons to know if a process is TS or DS (e.g. trend-cycle decomposition); can test this using (Augmented) Dickey-Fuller.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 5 23 / 23