ECON7350 Trends and Cycles
The University of Queensland
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Applied Econometrics for Macro and Finance
Many (not all) economic time series data appear to increase or decrease over time.
AUS Real GDP per Capita for 1973Q3:2019Q3
1980 1990 2000 2010 2020
Applied Econometrics for Macro and Finance
Real GDP per Capita 10000 16000
Trends in the Process Mean
A persistent increase or decrease in the data indicates a trend in the underlying stochastic process.
A trend in the process implies it is not stationary: either the mean or variance (or both) are changing over time.
Trends in the process mean are of theoretical interest in economics and finance. Decomposing the processes generating economic data into trends and cycles is
useful for policy.
Trends in macroeconomic processes are related to growth theory and capture long-term properties: e.g., potential GDP, trend inflation, natural rate of unemployment, etc.
Cycles are fluctuations around the trend and are related to business cycle theory.
Applied Econometrics for Macro and Finance
Deterministic Trends
Let {yt} be a non-stationary process decomposed as yt = ξt + ζt, where ξt is the trending (also called growth or secular) component;
ζt is the cyclical component.
{ζt} is a stationary process; it can be modelled by, e.g., an ARMA(p,q).
{ξt} can be modelled by a deterministic function of time, e.g., linear: ξt = ξ0 + ξ1t;
quadratic: ξt = ξ0 + ξ1t + ξ2t2;
any other function.
Such a specification is referred to as a deterministic trend model: the key assumption is that trend ξt and cycles ζt are independent: shocks to ζt do not affect ξt.
Applied Econometrics for Macro and Finance
Deterministic Trends in Practice
Modelling a deterministic trend in practice simply involves including time t (along with possibly t2, etc.) as a “regressor” in the model, e.g.
yt =a0 +δt+a1yt−1 +···+apyt−p +εt +b1εt−1 +···+bqεt−q. This is an ARMA(p, q) with a linear deterministic trend.
The implied trend-cycle decomposition is
ξt = a0 −δ/a(1)(a1 +2a2 +···pap)+ δ t, ζt = b(L)εt. a(1) a(1) a(L)
For business cycle analysis, ζt is of primary interest. Historically, ζt was extracted from yt by de-trending: regress yt on t and save the estimated residuals.
Applied Econometrics for Macro and Finance
Stochastic Trends
The assumption that trends and cycles are independent is problematic in economics. For example, a productivity shock has both short- and long-run effects on output.
Modelling economic time series with a deterministic trend leads to inaccurate inference: e.g., overestimate the persistence and variance of the business cycle.
An alternative approach to modelling the non-stationarity in the mean of {yt} is to assume that shocks have a permanent effect.
Let ξt = μ + ξt−1 + ηt, where {ηt} is white noise. Consequently, ξt = ξ0 + μt + η1 + . . . + ηt.
ξt is referred to as a stochastic trend.
Note that the model for ξt is a special case of an AR(1).
Applied Econometrics for Macro and Finance
Stochastic Trends and Random Walks
Recall that in the stable AR(1), we have:
yt =a0 +a1yt−1 +εt, |a1|<1,
=a0(1+a1 +···+at−1)+at1y0 +εt +a1εt−1 +···+at−1ε1, 11
so that the effect of a shock dies out as t increases; {yt} is stationary if Var(εt) = σε2. A stochastic trend can be viewed as being generated by the AR(1) with a1 = 1. Hence,
yt = a0 + yt−1 + εt,
=a0t+y0 +εt +εt−1 +···+ε1,
so that 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.
Applied Econometrics for Macro and Finance
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.
Applied Econometrics for Macro and Finance
Random Walk with a Drift
The AR(1) with a0 ̸= 0 and a1 = 1 is called a random walk with a drift. Hence, a stochastic trend is in fact a random walk with a drift.
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) = y0 + a0t,
Var(yt|y0)=Var(a0t+y0 +εt +···+ε1|y0)=tσε2,
Cov((yt, yt−s | y0) = (t − s)σε2. Therefore, all moments depend on t.
Applied Econometrics for Macro and Finance
Unit Root Processes
Recall that for the AR(1), the characteristic equation is a(z) = 1 − a1z = 0; therefore, 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 canberegardedasa
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.
Applied Econometrics for Macro and Finance
Example: AR(2)
For the model
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
a(z)= 1+a +a2+4a z 1+a −a2+4a z . 221 221
The necessary and sufficient conditions for stability are:
a1 +a2 <1, −a1 +a2 <1, −a2 <1.
Ifa1+a2 =1,thentheAR(2)hasaunitroot.
yt = a1yt−1 + a2yt−2 + εt,
Applied Econometrics for Macro and Finance
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 (homoscedastic) process is difference stationary or integrated of order one, written 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).
TheAR(2)witha1+a2 =1and|a2|<1isI(1).
Applied Econometrics for Macro and Finance
Beveridge- of I(1) Processes Letθ(L)=1+θ1L+θ2L2+··· withθ(1)=1+θ1+θ2+··· afiniteconstant.
If θ(L) = b(L)/a(L) and a(z) ̸= 0 for all |z| <= 1, then θ(1) is finite. 2
Thereisexistsaθ(L)=1+θ1L+θ2L +··· suchthatθ(L)=θ(1)+∆θ(L). If the non-stationary {yt} is I(1), then ∆yt is stationary and can be modelled as
∆yt = μ + θ(L)εt. Applying the decomposition of θ(L):
∆yt = μ + θ(1)εt + ∆θ(L)εt,
and letting ∆ξt = μ + θ(1)εt, ζt = θ(L)εt, yields ∆yt = ∆ξt + ∆ζt; hence, yt = ξt + ζt.
Applied Econometrics for Macro and Finance
Beveridge- of I(1) Processes This is the Beveridge-Nelson (BN) decomposition with:
thetrendξt beingaRWwithadrift: ξt =μ+ξt−1+ηt,ηt =θ(1)εt;
the cycles being ζt = θ(L)εt, where θ(1) ̸= 0 because {yt} is I(1).
The shock to the trend ηt is perfectly correlated with the shock to the cycles εt.
The trend ξt that is extracted by the BN decomposition can be interpreted as the optimal long-horizon conditional point forecast of the time series process {yt}, with any future drift removed (Morley, 2010, MD):
ξt= lim E(yt+h−μh|It). h−→∞
Applied Econometrics for Macro and Finance
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)?
Applied Econometrics for Macro and Finance
Augmented Dickey-
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
and test γ = 0.
∆yt = γyt−1 + b1∆yt−1 + · · · + bp−1∆yt−p+1 + εt,
Applied Econometrics for Macro and Finance
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.
Applied Econometrics for Macro and Finance
Augmented Dickey-
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.
interpretation: under H0, the process is I(1) with a deterministic trend (i.e., linear growth); under H1, it has a deterministic trend only.
The test statistic is the same as in a standard t-test.
The sampling distribution of the test statistic, however, is generally not standard: different critical values or p-values must be used.
The approach to working with ADF models is the same as with ARMA and ARDL models: the key assumption is that εt is mean-independent; specification uncertainty (choice of p and/or restrictions) must be accounted for.
Applied Econometrics for Macro and Finance
Variants of the ADF
There are many variants of the standard ADF specification: e.g., omit a0 and/or δt. ThestandardtestisstillH0 :γ=0againstH1 :γ<0.
The sampling distribution of the test statistic is generally different.
The interpretation of the null and alternative hypotheses is different.
Variants of the ADF can have very different power (prob. of rejecting a false H0).
Proposed changes to the standard ADF include joint null hypotheses.
These are mainly motivated by the quest for testing power, but may sometimes be sensible in a certain context.
Example: test for both a unit root and deterministic trend jointly. H0 :δ=0,γ=0againstH1 :δ̸=0,orγ<0,orδ̸=0,γ<0.
interpretation: under H0, the process is I(1) with constant growth; under H1, it is either stationary (δ = 0 and γ < 0) or has a deterministic trend only (δ ̸= 0 and γ < 0) or I(1) with linear growth (δ ̸= 0 and γ = 0).
Applied Econometrics for Macro and Finance
Trends in Practice
It is easy to get lost in the web of hypothesis testing when it comes to trend-searching!
The path to clarity is common sense.
The problem: conceptual vs empirical. They are not a perfect fit!
The solution: understanding where they are aligned and where discrepancies occur.
Applied Econometrics for Macro and Finance
Conceptual
Economic data often appears to be generated by a non-stationary process: trending behaviour in GDP per capita, wandering interest rates, etc.
The type of non-stationary process has important economic implications. Deterministic trends: predictable behaviour with no long-run memory.
Stochastic trends: increasing variance implying growing unpredictability in the future, permanent effect of shocks, complete long-run memory.
The importance of identifying the type of trend in practice depends on the application. Most economic processes are theoretically incompatible with a purely deterministic trend; and deterministic trend models do not fit economic data well. Conceptually, trending economic processes are better described by a combination of deterministic trends and unit roots.
Identifying unit roots is sometimes important, such as when the purpose of the empirical investigation requires it (e.g. trend-cycle decomposition).
Applied Econometrics for Macro and Finance
In practice, it is imperative to remember that we always work with realisations: we observe real-world data assumed to be realised from a stochastic process {yt}.
We reasonably assume {yt} is an ARMA(p,q) and proceed to narrow down an adequate set of models.
We are already aware that there is no such thing as a true model.
Any given model will have unknown parameters: in the simplest case of AR(1), the parameters are a0, a1 and σε2.
We examine the AR(1), but the same reasoning holds for all ARMAs.
We do not know the values of a0, a1, σε2; so we obtain realised AR(1)s empirically. We assume a0, a1 and σε2 can take on certain values; typically, a0 and a1 are any real values and σε2 any positive, real.
We then estimate a0, a1 and σε2 by choosing one value for each parameter from the respective sets, as functions of the observed data.
Applied Econometrics for Macro and Finance
Conceptual versus Empirical
Consideradatasety1,...,yT for(log)realGDPpercapita.
In the empirical setting, what are the chances that a1 is exactly 1 so that the process
generating this data is exactly I(1)?
A way to think about this (based on rigorous mathematics) is as follows.
Suppose we have specified an AR(1), but have not yet estimated any parameters. Consider “reasonable values” of a1 for GDP over the entire range of real numbers. What is Pr(a1 > 5)? Not likely, but many reasonable (small) probabilities.
What is Pr(0 < a1 ≤ 2)? More likely; many reasonable (large) probabilities.
What is Pr(0.99 < a1 ≤ 1.01)? many reasonable probabilities, even 1.
What is Pr(a1 = 1)? Zero.
By estimating a1, we infer the DGP empirically. However, there is a zero chance that any given inferred AR(1) is exactly a RW, although it may be close to one.
Applied Econometrics for Macro and Finance
Resolving the Conflict
H0 : γ = 0 represents a conceptual construct; an I(1) process exists only hypothetically. Reject H0: the inferred process is empirically distinguishable from a hypothetical unit
root process.
Fail to reject H0: the inferred process is not empirically distinguishable from a
hypothetical unit root process, although it is not exactly I(1).
The term not empirically distinguishable cannot be made more precise.
This is an ambiguity that is partially resolved with data: the more informative the data, the stronger is the meaning of not empirically distinguishable.
This ambiguity cannot be eliminated by altering hypothesis tests. This ambiguity must be accounted for in the inference!
Applied Econometrics for Macro and Finance
ADF Test Example: Australian Real GDP per Capita
Using the adf.test command provided by the aTSA package in R, we obtain the following output.
Augmented Dickey-
alternative: stationary
Type 1: no drift no trend Type 2: with drift no trend Type 3: with drift and trend
lag ADF p.value lag ADF p.value lag ADF p.value
[1,] 0 11.48 0.99 [1,] 0 1.0325 0.990 [1,] 0 -2.12 0.525
[2,] 1 2.74 0.99 [2,] 1 -0.0899 0.947 [2,] 1 -3.11 0.114
[3,] 2 6.89 0.99 [3,] 2 0.6191 0.990 [3,] 2 -1.84 0.643
[4,] 3 3.21 0.99 [4,] 3 -0.1421 0.939 [4,] 3 -2.21 0.485
[5,] 4 4.02 0.99 [5,] 4 0.1071 0.964 [5,] 4 -1.91 0.613
Note: in fact, p.value = 0.01 means p.value <= 0.01
Applied Econometrics for Macro and Finance
ADF Test Example: Australian Real GDP per Capita
Using the adf.test command provided by the aTSA package in R, we obtain the following output.
Augmented Dickey-
alternative: stationary
Type 1: no drift no trend Type 2: with drift no trend Type 3: with drift and trend
lag ADF p.value lag ADF p.value lag ADF p.value
[1,] 0 11.48 0.99 [1,] 0 1.0325 0.990 [1,] 0 -2.12 0.525
[2,] 1 2.74 0.99 [2,] 1 -0.0899 0.947 [2,] 1 -3.11 0.114
[3,] 2 6.89 0.99 [3,] 2 0.6191 0.990 [3,] 2 -1.84 0.643
[4,] 3 3.21 0.99 [4,] 3 -0.1421 0.939 [4,] 3 -2.21 0.485
[5,] 4 4.02 0.99 [5,] 4 0.1071 0.964 [5,] 4 -1.91 0.613
Note: in fact, p.value = 0.01 means p.value <= 0.01
Applied Econometrics for Macro and Finance
ADF Test Example: Australian Real GDP per Capita
Using the adf.test command provided by the aTSA package in R, we obtain the following output.
Augmented Dickey-
alternative: stationary
Type 1: no drift no trend Type 2: with drift no trend Type 3: with drift and trend
lag ADF p.value lag ADF p.value lag ADF p.value
[1,] 0 11.48 0.99 [1,] 0 1.0325 0.990 [1,] 0 -2.12 0.525
[2,] 1 2.74 0.99 [2,] 1 -0.0899 0.947 [2,] 1 -3.11 0.114
[3,] 2 6.89 0.99 [3,] 2 0.6191 0.990 [3,] 2 -1.84 0.643
[4,] 3 3.21 0.99 [4,] 3 -0.1421 0.939 [4,] 3 -2.21 0.485
[5,] 4 4.02 0.99 [5,] 4 0.1071 0.964 [5,] 4 -1.91 0.613
Note: in fact, p.value = 0.01 means p.value <= 0.01
Applied Econometrics for Macro and Finance
ADF Test Example: Australian Real GDP per Capita
Using the adf.test command provided by the aTSA package in R, we obtain the following output.
Augmented Dickey-
alternative: stationary
Type 1: no drift no trend Type 2: with drift no trend Type 3: with drift and trend
lag ADF p.value lag ADF p.value lag ADF p.value
[1,] 0 11.48 0.99 [1,] 0 1.0325 0.990 [1,] 0 -2.12 0.525
[2,] 1 2.74 0.99 [2,] 1 -0.0899 0.947 [2,] 1 -3.11 0.114
[3,] 2 6.89 0.99 [3,] 2 0.6191 0.990 [3,] 2 -1.84 0.643
[4,] 3 3.21 0.99 [4,] 3 -0.1421 0.939 [4,] 3 -2.21 0.485
[5,] 4 4.02 0.99 [5,] 4 0.1071 0.964 [5,] 4 -1.91 0.613
Note: in fact, p.value = 0.01 means p.value <= 0.01
Applied Econometrics for Macro and Finance
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